Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Humaira Yasmin; Aljawhara H. Almuqrin; Humaira Yasmin; Aljawhara H. Almuqrin

doi:10.3934/math.2024811

AIMS Mathematics

2024, Volume 9, Issue 6: 16721-16752. doi: 10.3934/math.2024811

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Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies

Humaira Yasmin ^{1,2
,
,},
Aljawhara H. Almuqrin ³

1.
Department of Basic Sciences, General Administration of Preparatory Year, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al Ahsa 31982, Saudi Arabia
3.
Department of Physics, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia

Received: 24 March 2024 Revised: 02 May 2024 Accepted: 07 May 2024 Published: 14 May 2024
MSC : 34G20, 35A20, 35A22, 35R11

Within the framework of time fractional calculus using the Caputo operator, the Aboodh residual power series method and the Aboodh transform iterative method were implemented to analyze three basic equations in mathematical physics: the heat equation, the diffusion equation, and Burger's equation. We investigated the analytical solutions of these equations using Aboodh techniques, which provide practical and precise methods for solving fractional differential equations. We clarified the behavior and properties of the obtained approximations using the suggested methods through exact mathematical derivations and computational analysis. The obtained approximations were analyzed numerically and graphically to verify their high accuracy and stability against different related parameters. Additionally, we examined the impact of varying the fractional parameter the profiles of all derived approximations. Our results confirm these methods, efficacy in capturing the complicated dynamics of fractional systems. Therefore, they enhance the comprehension and examination of time-fractional equations in many scientific and technical contexts and in modeling different physical problems related to fluid mediums and plasma physics.

Keywords:

Citation: Humaira Yasmin, Aljawhara H. Almuqrin. Analytical study of time-fractional heat, diffusion, and Burger's equations using Aboodh residual power series and transform iterative methodologies[J]. AIMS Mathematics, 2024, 9(6): 16721-16752. doi: 10.3934/math.2024811

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Abstract

1. Introduction

Hermite-Hadamard inequality is a double inequality for convex functions that has a lot of literary value (please see ^[16]).

Let $\mathbb{\zeta}:I\longrightarrow \mathbb{R},$ $\emptyset\neq I\subseteq \mathbb{R}$ , $\mathfrak{\varsigma}, \mathbb{\tau}\in I$ with $\mathfrak{\varsigma } < \mathbb{\tau}$ , be a convex function. Then

$\begin{equation} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) \leq\frac{1}{\mathbb{\tau}-\mathfrak{\varsigma}} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}(\mathfrak{w})d\mathfrak{w}\leq\frac{\mathbb{\zeta }(\mathfrak{\varsigma})+\mathbb{\zeta}(\mathbb{\tau})}{2}, \end{equation}$

(1.1)

the inequality holds in reversed direction if $\mathbb{\zeta}$ is concave.

Fejér ^[15] established the following double inequality as a weighted generalization of (1.1):

$\begin{equation} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\vartheta}(\mathfrak{w})d\mathfrak{w}\leq { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}(\mathfrak{w})\mathbb{\vartheta}(\mathfrak{w})d\mathfrak{w} \leq\frac{\mathbb{\zeta}(\mathfrak{\varsigma})+\mathbb{\zeta}(\mathbb{\tau} )}{2} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\vartheta}(\mathfrak{w})d\mathfrak{w}, \end{equation}$

(1.2)

where $\mathbb{\zeta}:I\longrightarrow \mathbb{R},$ $\emptyset\neq I\subseteq \mathbb{R}$ , $\mathfrak{\varsigma}, \mathbb{\tau}\in I$ with $\mathfrak{\varsigma } < \mathbb{\tau}$ is any convex function and $\mathbb{\vartheta}:\left[\mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is non-negative integrable and symmetric with respect to $\mathfrak{w} = \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}$ .

These inequalities have many extensions and generalizations, see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]} and ^{[19,20,21,22,23,24,25,26,27,28,29,30,31]}.

Consider the following mappings on $\left[ 0, 1\right]$ :

$\mathcal{\breve{G}}(\mathfrak{\iota}) = \frac{1}{2} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \left[ \mathbb{\zeta}\left( \mathfrak{\iota\varsigma}+\left( 1-\mathfrak{\iota}\right) \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2}\right) +\mathbb{\zeta}\left( \mathfrak{\iota}\mathbb{\tau}+\left( 1-\mathfrak{\iota}\right) \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2}\right) \right] d\mathfrak{w},$

$\mathbb{H}(\mathfrak{\iota}) = \frac{1}{\mathbb{\tau}-\mathfrak{\varsigma}} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) d\mathfrak{w},$

$\mathbb{H}_{\mathbb{\vartheta}}(\mathfrak{\iota}) = \frac{1}{\mathbb{\tau }-\mathfrak{\varsigma}} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) d\mathfrak{w},$

$\mathcal{L}(\mathfrak{\iota}) = \frac{1}{2\left( \mathbb{\tau} -\mathfrak{\varsigma}\right) } { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \left[ \mathbb{\zeta}\left( \mathfrak{\iota\varsigma}+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\right) +\mathbb{\zeta}\left( \mathfrak{\iota}\mathbb{\tau}+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\right) \right] d\mathfrak{w}$

and

$\mathcal{L}_{\mathbb{\vartheta}}(\mathfrak{\iota}) = \frac{1}{2} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \left[ \mathbb{\zeta}\left( \mathfrak{\iota\varsigma}+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\right) +\mathbb{\zeta}\left( \mathfrak{\iota}\mathbb{\tau}+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\right) \right] \mathbb{\vartheta}(\mathfrak{w})d\mathfrak{w},$

where $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is a convex function and $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma }, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is non-negative integrable and symmetric with respect to $\mathfrak{w} = \frac {\mathfrak{\varsigma}+\mathbb{\tau}}{2}$ .

The important results that characterize the properties of the above mappings and inequalities are discussed by a number of mathematicians.

Dragomir ^[2] established the theorem which refines the first inequality of (1.1).

Theorem 1. ^[2] Let $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ be a convex function on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ . Then $\mathbb{H}$ is monotonically increasing and convex on $\left[ 0, 1\right]$ . Moreover, one has thefollowing inequalities

$\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) = \mathbb{H}\left( 0\right) \leq\mathbb{H}\left( \mathfrak{\iota}\right) \leq\mathbb{H}\left( 1\right) = \frac{1}{\mathbb{\tau}-\mathfrak{\varsigma} }\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w}.$

Dragomir et al. ^[7] obtained the refinements of (1.1).

Theorem 2. ^[2] Let $\mathbb{\zeta}$ , $\mathbb{H}$ be defined asabove. Then

(i) The following inequality holds

$\begin{align*} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) & \leq\frac{2}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\frac {3\mathfrak{\varsigma}+\mathbb{\tau}}{4}}^{\frac{\mathfrak{\varsigma }+3\mathbb{\tau}}{4}}\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w} \leq\int_{0}^{1}\mathbb{H}\left( \mathfrak{\iota}\right) d\mathfrak{\iota}\\ & \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2}\right) +\frac{1}{\mathbb{\tau}-\mathfrak{\varsigma}} \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w}\right] . \end{align*}$

(ii) If $\mathbb{\zeta}$ is differentiable on $\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ , one has

$\begin{align*} 0 & \leq\frac{1}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma }}^{\mathbb{\tau}}\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w} -\mathbb{H}\left( \mathfrak{\iota}\right) \\ & \leq\left( 1-\mathfrak{\iota}\right) \left[ \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2}-\frac{1}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w} \right] \end{align*}$

and

$0\leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}-\mathbb{H}\left( \mathfrak{\iota}\right) \leq\frac{\left( \mathbb{\zeta}^{^{\prime}}\left( \mathbb{\tau}\right) -\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{\varsigma}\right) \right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) }{4}.$

Theorem 3. ^[7] Let $\mathbb{\zeta}$ , $\mathbb{H}$ , $\mathcal{\breve{G}}$ be defined as above. We have

(i) $\mathcal{\breve{G}}$ is convex and increasing on $\left[0, 1\right]$ .

(ii) The following hold

$\underset{\mathfrak{\iota}\in\left[ 0,1\right] }{\inf}\mathcal{\breve{G} }\left( \mathfrak{\iota}\right) = \mathcal{\breve{G}}\left( 0\right) = \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right)$

and

$\underset{\mathfrak{\iota}\in\left[ 0,1\right] }{\sup}\mathcal{\breve{G} }\left( \mathfrak{\iota}\right) = \mathcal{\breve{G}}\left( 1\right) = \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}.$

(iii) The inequality

$\mathbb{H}\left( \mathfrak{\iota}\right) \leq\mathcal{\breve{G}}\left( \mathfrak{\iota}\right)$

holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

(iv) Then

$\begin{align*} \frac{2}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\frac{3\mathfrak{\varsigma }+\mathbb{\tau}}{4}}^{\frac{\mathfrak{\varsigma}+3\mathbb{\tau}}{4} }\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w} & \leq\frac{1} {2}\left[ \mathbb{\zeta}\left( \frac{3\mathfrak{\varsigma}+\mathbb{\tau}} {4}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+3\mathbb{\tau} }{4}\right) \right] \\ & \leq\int_{0}^{1}\mathcal{\breve{G}}\left( \mathfrak{\iota}\right) d\mathfrak{\iota}\\ & \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2}\right) +\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma }\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}\right] . \end{align*}$

(v) If $\mathbb{\zeta}$ is differentiable on $\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ , one has

$0\leq\mathbb{H}\left( \mathfrak{\iota}\right) -\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) \leq\mathcal{\breve{G} }\left( \mathfrak{\iota}\right) -\mathbb{H}\left( \mathfrak{\iota}\right) .$

Theorem 4. ^[7] Let $\mathbb{\zeta}$ , $\mathbb{H}$ , $\mathcal{\breve{G}}$ , $\mathcal{L}$ be defined as above. Then

(i) $\mathcal{L}$ is convex on $\left[ 0, 1\right]$ .

(ii) The inequalities

$\mathcal{\breve{G}}\left( \mathfrak{\iota}\right) \leq\mathcal{L}\left( \mathfrak{\iota}\right) \leq\frac{1-\mathfrak{\iota}}{\mathbb{\tau }-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\mathbb{\zeta}\left( \mathfrak{w}\right) d\mathfrak{w}+\mathfrak{\iota }\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}\leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}$

hold for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ and

$\underset{\mathfrak{\iota}\in\left[ 0,1\right] }{\sup}\mathcal{L}\left( \mathfrak{\iota}\right) = \mathcal{L}\left( 1\right) = \frac{\mathbb{\zeta }\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau }\right) }{2}.$

(iii) The inequalities

$\mathbb{H}\left( 1-\mathfrak{\iota}\right) \leq\mathcal{L}\left( \mathfrak{\iota}\right) \;{{and}}\;\frac{\mathbb{H}\left( \mathfrak{\iota }\right) +\mathbb{H}\left( 1-\mathfrak{\iota}\right) }{2}\leq \mathcal{L}\left( \mathfrak{\iota}\right)$

hold for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

Teseng et al. ^[24] proved the following result.

Lemma 1. ^[24] Let $\mathbb{\zeta}:$ $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ be a convex function and let $\mathfrak{\varsigma}\leq\breve{\varkappa}_{1}\leq\mathfrak{w}_{1}\leq\mathfrak{w}_{2}\leq\breve{\varkappa}_{2}\leq\mathbb{\tau}$ with $\mathfrak{w}_{1}+\mathfrak{w}_{2} = \breve{\varkappa}_{1}+\breve{\varkappa}_{2}$ . Then

$\mathbb{\zeta}\left( \mathfrak{w}_{1}\right) +\mathbb{\zeta}\left( \mathfrak{w}_{2}\right) \leq\mathbb{\zeta}\left( \breve{\varkappa} _{1}\right) +\mathbb{\zeta}\left( \breve{\varkappa}_{2}\right) .$

Yang and Tseng ^[28] proven the theorem by using Lemma 1 which refines the first inequality of (1.2) and generalizes Theorem 1.

Theorem 5. ^[28] Let $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ be a convex function and $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is non-negative integrable and symmetric with respect to $\mathfrak{w} = \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}$ . Then $\mathbb{H}_{\mathbb{\vartheta}}$ is convex, increasing on $\left[ 0, 1\right]$ , and for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ , we have

$\begin{equation} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2}\right) \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\mathbb{\vartheta}\left( \mathfrak{w}\right) d\mathfrak{w} = \mathbb{H}_{\mathbb{\vartheta}}\left( 0\right) \leq\mathbb{H}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \leq\mathbb{H}_{\mathbb{\vartheta}}\left( 1\right) = \int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\mathbb{\zeta}\left( \mathfrak{w} \right) \mathbb{\vartheta}\left( \mathfrak{w}\right) d\mathfrak{w}. \end{equation}$

(1.3)

One of the generalizations of the convex functions is harmonic functions:

Definition 1. ^[17] Define $I\subseteq \mathbb{R} \backslash\left\{ 0\right\}$ as an interval of real numbers. A function $\mathbb{\zeta}$ from $I$ to the real numbers is considered to be harmonically convex, if

$\begin{equation} \mathbb{\zeta}\left( \frac{\mathfrak{w}\breve{\varkappa}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \breve{\varkappa}}\right) \leq\mathfrak{\iota}\mathbb{\zeta}\left( \breve{\varkappa}\right) +\left( 1-\mathfrak{\iota}\right) \mathbb{\zeta}\left( \mathfrak{w}\right) \end{equation}$

(1.4)

for all $\mathfrak{w}, \breve{\varkappa}\in I$ and $\mathfrak{\iota}\in\left[ 0, 1\right]$ . Harmonically concave $\mathbb{\zeta}$ is defined as the inequality in (1.4) reversed.

İşcan ^[17] used harmonic-convexity to develop the inequalities of Hermite-Hadamard type.

Theorem 6. ^[17] Let $\mathbb{\zeta}:I\subseteq \mathbb{R} \backslash\left\{ 0\right\} \rightarrow \mathbb{R}$ be a harmonically convex function and $\mathfrak{\varsigma}, \mathbb{\tau}\in I$ with $\mathfrak{\varsigma} < \mathbb{\tau}$ . If $\mathbb{\zeta}\in L\left(\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \right)$ then theinequalities

$\begin{equation} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \leq\frac{\mathfrak{\varsigma }\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathbb{\tau} }^{\mathfrak{\varsigma}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2} \end{equation}$

(1.5)

hold.

Let $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset\left(0, \infty\right) \rightarrow \mathbb{R}$ be a harmonic convex mapping and let $\mathbb{S}, \mathbb{V}:\left[ 0, 1\right] \rightarrow \mathbb{R}$ be defined by

$\begin{equation} \mathbb{S}\left( \mathfrak{\iota}\right) = \frac{\mathfrak{\varsigma }\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{1}{\mathfrak{w}^{2}}\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) d\mathfrak{w} \end{equation}$

(1.6)

and

$\begin{array}{l} \mathbb{V}\left( \mathfrak{\iota}\right) = \frac{\mathfrak{\varsigma }\mathbb{\tau}}{2\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathbb{\tau}\mathfrak{w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) \right. \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) \right] d\mathfrak{w}. \end{array}$

(1.7)

The author obtained the refinement inequalities for (1.5) related to the above mappings:

Theorem 7. ^[21] Let $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset\left(0, \infty\right) \rightarrow \mathbb{R}$ be a harmonic convex function on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ . Then

(i) $\mathbb{S}$ is harmonic convex $\left(0, 1\right]$ andincreases monotonically on $\left[ 0, 1\right]$ .

(ii) The following hold:

$\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) = \mathbb{S}\left( 0\right) \leq\mathbb{S}\left( \mathfrak{\iota}\right) \leq\mathbb{S}\left( 1\right) = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}.$

Theorem 8. ^[21] Let $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset\left(0, \infty\right) \rightarrow \mathbb{R}$ be a harmonic convex function on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ . Then

(i) $\mathbb{V}$ is harmonic convex $\left(0, 1\right]$ andincreases monotonically on $\left[ 0, 1\right]$ .

(ii) The following hold:

$\frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} = \mathbb{V}\left( 0\right) \leq\mathbb{V}\left( \mathfrak{\iota}\right) \leq\mathbb{V}\left( 1\right) = \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}.$

Harmonic symmetricity of a function is given in the definition below.

Definition 2. ^[22] A function $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma }, \mathbb{\tau}\right] \subseteq \mathbb{R} \backslash\left\{ 0\right\} \rightarrow \mathbb{R}$ is harmonically symmetric with respect to $\frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}$ if

$\mathbb{\vartheta}\left( \mathfrak{w}\right) = \mathbb{\vartheta}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\frac {1}{\mathfrak{w}}}\right)$

holds for all $\mathfrak{w}\in\left[ \mathfrak{\varsigma}, \mathbb{\tau }\right]$ .

Fejér type inequalities using harmonic convexity and the notion of harmonic symmetricity were presented in Chan and Wu ^[1].

Theorem 9. ^[1] Let $\mathbb{\zeta}:I\subseteq \mathbb{R} \backslash\left\{ 0\right\} \rightarrow \mathbb{R}$ be a harmonically convex function and $\mathfrak{\varsigma}, \mathbb{\tau}\in I$ with $\mathfrak{\varsigma} < \mathbb{\tau}$ . If $\mathbb{\zeta}\in L\left(\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \right)$ and $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]\subseteq \mathbb{R} \backslash\left\{ 0\right\} \rightarrow \mathbb{R}$ is nonnegative, integrable and harmonically symmetric with respect to $\frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}$ , then

$\begin{equation} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int_{\mathbb{\tau} }^{\mathfrak{\varsigma}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\int_{\mathbb{\tau}}^{\mathfrak{\varsigma }}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\frac{\mathbb{\zeta }\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau }\right) }{2}\int_{\mathbb{\tau}}^{\mathfrak{\varsigma}}\frac {\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}. \end{equation}$

(1.8)

Chan and Wu ^[1] also defined some mappings related to (1.8) and discussed important properties of these mappings.

Motivated by the studies conducted in ^[2,21,24,27], we define some new mappings in connection to (1.8) and to prove new Féjer type inequalities which indeed provide refinement inequalities as well.

2. Main results

To prove the major findings of this work, we employ the given important facts about harmonic convex and convex functions.

Theorem 10. ^[8,9] If $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset I\subset\left(0, \infty\right)$ and if we consider thefunction $\breve{h}:\left[ \frac{1}{\mathbb{\tau}}, \frac{1}{\mathfrak{\varsigma}}\right] \rightarrow \mathbb{R}$ defined by $\breve{h}\left(\mathfrak{\iota}\right) = \mathbb{\zeta}\left(\frac{1}{\mathfrak{\iota}}\right)$ , then $\mathbb{\zeta}$ is harmonicallyconvex on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ if and only if $\breve{h}$ is convex in the usual sense on $\left[ \frac{1}{\mathbb{\tau}}, \frac{1}{\mathfrak{\varsigma}}\right]$ .

Theorem 11. ^[8,9] If $I\subset\left(0, \infty\right)$ and $\mathbb{\zeta}$ is convex and nondecreasing function then $\mathbb{\zeta}$ isharmonic convex and if $\mathbb{\zeta}$ is harmonic convex and nonincreasing function then $\mathbb{\zeta}$ is convex.

Let us now define some mappings on $\left[ 0, 1\right]$ related to (1.8) and prove some refinement inequalities.

$\mathcal{\breve{G}}_{1}(\mathfrak{\iota}) = \frac{1}{2}\left[ \mathbb{\zeta }\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathfrak{\varsigma\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) }\right) +\mathbb{\zeta}\left( \frac {2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \right] d\mathfrak{w},$

$\mathbb{S}(\mathfrak{\iota}) = \frac{\mathfrak{\varsigma}\mathbb{\tau} }{\mathbb{\tau}-\mathfrak{\varsigma}} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \frac{1}{\mathfrak{w}^{2}}\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) d\mathfrak{w},$

$\mathbb{S}_{\mathbb{\vartheta}}(\mathfrak{\iota}) = { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w} }\right) \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) } {\mathfrak{w}^{2}}d\mathfrak{w},$

$\mathbb{T}(\mathfrak{\iota}) = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{2\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{\mathbb{\tau }\mathfrak{w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma w} }{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma} }\right) \right] d\mathfrak{w}$

and

$\mathbb{T}_{\mathbb{\vartheta}}(\mathfrak{\iota}) = \frac{1}{2} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \left[ \mathbb{\zeta}\left( \frac{\mathbb{\tau}\mathfrak{w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) \right] \frac{\mathbb{\vartheta}(\mathfrak{w})}{\mathfrak{w}^{2}}d\mathfrak{w},$

where $\mathbb{\zeta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is a harmonic convex function and $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is non-negative integrable and symmetric with respect to $\mathfrak{w} = \frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}$ .

Lemma 2. Let $\mathbb{\zeta}:$ $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset\left(0, \infty\right) \rightarrow \mathbb{R}$ be a harmonic convex function and let $\mathfrak{\varsigma}\leq\breve{\varkappa}_{1}\leq\mathfrak{w}_{1}\leq\mathfrak{w}_{2}\leq\breve{\varkappa}_{2}\leq\mathbb{\tau}$ with $\frac{\mathfrak{w}_{1}\mathfrak{w}_{2}}{\mathfrak{w}_{1}+\mathfrak{w}_{2}} = \frac{\breve{\varkappa}_{1}\breve{\varkappa}_{2}}{\breve{\varkappa}_{1}+\breve{\varkappa}_{2}}$ . Then

Proof. For $\breve{\varkappa}_{1} = \breve{\varkappa}_{2}$ , the result is obvious. We observe that

$\mathfrak{w}_{1} = \frac{\breve{\varkappa}_{1}\breve{\varkappa}_{2}}{\left( \frac{\mathfrak{w}_{1}\breve{\varkappa}_{1}-\breve{\varkappa}_{1} \breve{\varkappa}_{2}}{\mathfrak{w}_{1}\breve{\varkappa}_{1}-\mathfrak{w} _{1}\breve{\varkappa}_{2}}\right) \breve{\varkappa}_{2}+\left( \frac {\breve{\varkappa}_{1}\breve{\varkappa}_{2}-\mathfrak{w}_{1}\breve{\varkappa }_{2}}{\mathfrak{w}_{1}\breve{\varkappa}_{1}-\mathfrak{w}_{1}\breve{\varkappa }_{2}}\right) \breve{\varkappa}_{1}}\;{\rm{ and }}\;\mathfrak{w}_{2} = \frac {\breve{\varkappa}_{1}\breve{\varkappa}_{2}}{\left( \frac{\mathfrak{w} _{2}\breve{\varkappa}_{1}-\breve{\varkappa}_{1}\breve{\varkappa}_{2} }{\mathfrak{w}_{2}\breve{\varkappa}_{1}-\mathfrak{w}_{2}\breve{\varkappa}_{2} }\right) \breve{\varkappa}_{2}+\left( \frac{\breve{\varkappa}_{1} \breve{\varkappa}_{2}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}{\mathfrak{w} _{2}\breve{\varkappa}_{1}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}\right) \breve{\varkappa}_{1}}.$

By applying the harmonic convexity, we obtain

$\begin{align*} \mathbb{\zeta}\left( \mathfrak{w}_{1}\right) +\mathbb{\zeta}\left( \mathfrak{w}_{2}\right) & \leq\left( \frac{\mathfrak{w}_{1}\breve {\varkappa}_{1}-\breve{\varkappa}_{1}\breve{\varkappa}_{2}}{\mathfrak{w} _{1}\breve{\varkappa}_{1}-\mathfrak{w}_{1}\breve{\varkappa}_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{1}\right) +\left( \frac {\breve{\varkappa}_{1}\breve{\varkappa}_{2}-\mathfrak{w}_{1}\breve{\varkappa }_{2}}{\mathfrak{w}_{1}\breve{\varkappa}_{1}-\mathfrak{w}_{1}\breve{\varkappa }_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{2}\right) \\ & +\left( \frac{\mathfrak{w}_{2}\breve{\varkappa}_{1}-\breve{\varkappa} _{1}\breve{\varkappa}_{2}}{\mathfrak{w}_{2}\breve{\varkappa}_{1} -\mathfrak{w}_{2}\breve{\varkappa}_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{1}\right) +\left( \frac{\breve{\varkappa}_{1} \breve{\varkappa}_{2}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}{\mathfrak{w} _{2}\breve{\varkappa}_{1}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{2}\right) \\ & = \left( \frac{\mathfrak{w}_{1}\breve{\varkappa}_{1}-\breve{\varkappa} _{1}\breve{\varkappa}_{2}}{\mathfrak{w}_{1}\breve{\varkappa}_{1} -\mathfrak{w}_{1}\breve{\varkappa}_{2}}+\frac{\mathfrak{w}_{2}\breve {\varkappa}_{1}-\breve{\varkappa}_{1}\breve{\varkappa}_{2}}{\mathfrak{w} _{2}\breve{\varkappa}_{1}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{1}\right) \\ & +\left( \frac{\breve{\varkappa}_{1}\breve{\varkappa}_{2}-\mathfrak{w} _{1}\breve{\varkappa}_{2}}{\mathfrak{w}_{1}\breve{\varkappa}_{1} -\mathfrak{w}_{1}\breve{\varkappa}_{2}}+\frac{\breve{\varkappa}_{1} \breve{\varkappa}_{2}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}{\mathfrak{w} _{2}\breve{\varkappa}_{1}-\mathfrak{w}_{2}\breve{\varkappa}_{2}}\right) \mathbb{\zeta}\left( \breve{\varkappa}_{2}\right) \\ & = \frac{\breve{\varkappa}_{1}}{\breve{\varkappa}_{2}-\breve{\varkappa}_{1} }\left( \left( \frac{\mathfrak{w}_{1}+\mathfrak{w}_{2}}{\mathfrak{w} _{1}\mathfrak{w}_{2}}\right) \breve{\varkappa}_{2}-2\right) \mathbb{\zeta }\left( \breve{\varkappa}_{1}\right) \\ & +\frac{\breve{\varkappa}_{2}}{\breve{\varkappa}_{2}-\breve{\varkappa}_{1} }\left( 2-\left( \frac{\mathfrak{w}_{1}+\mathfrak{w}_{2}}{\mathfrak{w} _{1}\mathfrak{w}_{2}}\right) \breve{\varkappa}_{1}\right) \mathbb{\zeta }\left( \breve{\varkappa}_{2}\right) = \mathbb{\zeta}\left( \breve {\varkappa}_{1}\right) +\mathbb{\zeta}\left( \breve{\varkappa}_{2}\right) . \end{align*}$

We first prove a result similar to (1.3) for harmonically convex functions which provide refinement inequalities for (1.8).

Theorem 12. Let $\mathbb{\zeta}:$ $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \subset\left(0, \infty\right) \rightarrow \mathbb{R}$ be a harmonic convex function, $0 < \mathbb{\rho} < 1$ , $0 < \mathbb{\theta} < 1$ , $\mathbb{\lambda} = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho}\mathfrak{\varsigma}+\left(1-\mathbb{\rho}\right) \mathbb{\tau}}$ , $\mathbb{\tau}_{0} = \left(\frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\right) \min\left\{ \frac{\mathbb{\rho}}{1-\mathbb{\theta}}, \frac{1-\mathbb{\rho}}{\mathbb{\theta}}\right\}$ andlet $\mathbb{\vartheta}:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]\rightarrow \mathbb{R}$ be nonnegative and integrable and $\mathbb{\vartheta}\left(\frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota}\mathbb{\lambda}}\right) = \mathbb{\vartheta}\left(\frac{\mathbb{\lambda}}{1+\left(1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda}}\right)$ , $\mathfrak{\iota}\in\left[ 0, \mathbb{\tau}_{0}\right]$ . Then

$\begin{array}{l} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho }\mathfrak{\varsigma}+\left( 1-\mathbb{\rho}\right) \mathbb{\tau}}\right) \int_{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda}}}^{\frac{\mathbb{\lambda}}{1-\mathbb{\theta }\mathfrak{\iota}\mathbb{\lambda}}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\frac{1-\mathbb{\theta}}{\mathbb{\theta}}\int_{\frac{\mathbb{\lambda} }{1-\mathbb{\theta}\mathfrak{\iota}\mathbb{\lambda}}}^{\mathbb{\lambda}} \frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\frac{\mathbb{\theta} }{1-\mathbb{\theta}}\int_{\mathbb{\lambda}}^{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda}}}\frac {\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\\ \leq\left[ \mathbb{\rho\zeta}\left( \mathbb{\tau}\right) +\left( 1-\mathbb{\rho}\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) \right] \int_{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda}}}^{\frac{\mathbb{\lambda}}{1-\mathbb{\theta }\mathfrak{\iota}\mathbb{\lambda}}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.1)

Proof. For every $\mathbb{\tau}\in\left[ 0, \mathbb{\tau}_{0}\right]$ , we have the identity

$\begin{array}{l} \int_{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda}}}^{\frac{\mathbb{\lambda}}{1-\mathbb{\theta }\mathfrak{\iota}\mathbb{\lambda}}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} = \int_{\frac {\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota }\mathbb{\lambda}}}^{\mathbb{\lambda}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\int_{\mathbb{\lambda} }^{\frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota}\mathbb{\lambda}} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} \\ = \mathbb{\theta}\int_{0}^{\mathfrak{\iota}}\frac{\mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\left( 1-\mathbb{\theta}\right) \int _{0}^{\mathfrak{\iota}}\frac{\mathbb{\vartheta}\left( \frac{\mathbb{\lambda} }{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2} }d\mathfrak{w} = \int_{0}^{\mathfrak{\iota}}\frac{\mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.2)

We now prove that the mapping $\mathcal{W}:\left[ 0, \mathbb{\tau}_{0}\right] \rightarrow \mathbb{R}$ defined by

$\mathcal{W}\left( \mathfrak{\iota}\right) = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota }\mathbb{\lambda}}\right) +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda }}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda} }\right)$

is harmonic convex $\left(0, \mathbb{\tau}_{0}\right]$ and monotonically increasing on $\left[ 0, \mathbb{\tau}_{0}\right]$ .

Since the sum of two harmonic convex functions is a harmonic convex, hence $\mathcal{W}$ is a harmonic convex on $\left(0, \mathbb{\tau}_{0}\right]$ . Let $\mathfrak{\iota}\in\left(0, \mathbb{\tau}_{0}\right]$ , it follows from the harmonic convexity of $\mathbb{\zeta}$ that

(2.3)

We observed that $0 < \mathbb{\rho}\leq\frac{\mathbb{\rho}\left(\mathbb{\tau }-\mathfrak{\varsigma}\right) +\mathfrak{\varsigma}\mathbb{\tau\theta }\mathfrak{\iota}}{\mathbb{\tau}-\mathfrak{\varsigma}}\leq1$ , $0\leq \frac{\left(1-\mathbb{\rho}\right) \left(\mathbb{\tau} -\mathfrak{\varsigma}\right) -\mathbb{\theta}\mathfrak{\iota\varsigma }\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\leq1-\mathbb{\rho} < 1$ , $0\leq\mathbb{\rho}\leq\frac{\mathbb{\rho}\left(\mathbb{\tau} -\mathfrak{\varsigma}\right) -\left(1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\leq\mathbb{\rho}\leq1$ and $0 < 1-\mathbb{\rho}\leq\frac{\left(1-\mathbb{\rho}\right) \left(\mathbb{\tau}-\mathfrak{\varsigma}\right) +\left(1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau} }{\mathbb{\tau}-\mathfrak{\varsigma}}\leq1$ . Thus, by using the harmonic convexity, we obtain

$\begin{array}{l} \mathcal{W}\left( \mathfrak{\iota}\right) = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota }\mathbb{\lambda}}\right) +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda }}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda} }\right) \\ = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac {\mathfrak{\varsigma}\mathbb{\tau}}{\left( \frac{\mathbb{\rho}\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) +\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{\iota}\mathbb{\theta}}{\mathbb{\tau}-\mathfrak{\varsigma}}\;\;\right) \mathfrak{\varsigma}+\left( \frac{\left( 1-\mathbb{\rho}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) -\mathbb{\theta}\mathfrak{\iota \varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\;\;\right) \mathbb{\tau}}\right) \\ +\mathbb{\theta\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( \frac{\mathbb{\rho}\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) -\left( 1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau}}{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) }\;\;\right) \mathfrak{\varsigma }+\left( \frac{\left( 1-\mathbb{\rho}\right) \left( \mathbb{\tau }-\mathfrak{\varsigma}\right) +\left( 1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\;\;\;\right) \mathbb{\tau}}\right) \\ \leq\left( 1-\mathbb{\theta}\right) \left( \frac{\left( 1-\mathbb{\rho }\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) -\mathbb{\theta }\mathfrak{\iota\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\left( 1-\mathbb{\theta}\right) \\ \times\left( \frac{\mathbb{\rho}\left( \mathbb{\tau}-\mathfrak{\varsigma }\right) +\mathfrak{\iota\varsigma}\mathbb{\tau\theta}}{\mathbb{\tau }-\mathfrak{\varsigma}}\right) \mathbb{\zeta}\left( \mathbb{\tau}\right) +\mathbb{\theta}\left( \frac{\left( 1-\mathbb{\rho}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) +\left( 1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau}}{\left( \mathbb{\tau} -\mathfrak{\varsigma}\right) }\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) \\ +\mathbb{\theta}\left( \frac{\mathbb{\rho}\left( \mathbb{\tau} -\mathfrak{\varsigma}\right) -\left( 1-\mathbb{\theta}\right) \mathfrak{\iota\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\right) \mathbb{\zeta}\left( \mathbb{\tau}\right) = \left( 1-\mathbb{\rho }\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\rho \zeta}\left( \mathbb{\tau}\right) . \end{array}$

(2.4)

From (2.3) and (2.4), we obtain

$\begin{equation} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho }\mathfrak{\varsigma}+\left( 1-\mathbb{\rho}\right) \mathbb{\tau}}\right) \leq\mathcal{W}\left( \mathfrak{\iota}\right) \leq\left( 1-\mathbb{\rho }\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\rho \zeta}\left( \mathbb{\tau}\right) . \end{equation}$

(2.5)

Finally, for $\mathfrak{\iota}_{1}$ , $\mathfrak{\iota}_{2}$ , such that $0 < \mathfrak{\iota}_{1} < \mathfrak{\iota}_{2}\leq\mathfrak{\varsigma}_{0}$ , since $\mathcal{W}\left(\mathfrak{\iota}\right)$ is harmonic convex, it follows from (2.3) that

$\frac{\mathcal{W}\left( \mathfrak{\iota}_{2}\right) -\mathcal{W}\left( \mathfrak{\iota}_{1}\right) }{\mathfrak{\iota}_{2}-\mathfrak{\iota}_{1}} \geq0.$

This shows that $\mathcal{W}$ is increasing on $\left[ 0, \mathbb{\tau} _{0}\right]$ .

Since $\mathbb{\vartheta}$ is nonnegative, multiplying (2.5) by $\frac{\mathbb{\vartheta}\left(\frac{\mathbb{\lambda} }{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\right) }{\mathfrak{w}^{2}}$ , integrating the resulting inequalities over $\left[ 0, \mathfrak{\iota }\right]$ and using $\mathbb{\vartheta}\left(\frac{\mathbb{\lambda} }{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\right) = \mathbb{\vartheta }\left(\frac{\mathbb{\lambda}}{1+\left(1-\mathbb{\theta}\right) \mathfrak{w}\mathbb{\lambda}}\right)$ , we have

$\begin{array}{l} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho }\mathfrak{\varsigma}+\left( 1-\mathbb{\rho}\right) \mathbb{\tau}}\right) \int_{0}^{\mathfrak{\iota}}\frac{\mathbb{\vartheta}\left( \frac {\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\left( 1-\mathbb{\theta}\right) \int_{0}^{\mathfrak{\iota}} \frac{\mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta }\mathfrak{w}\mathbb{\lambda}}\;\;\right) \mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\mathbb{\theta}\int_{0}^{\mathfrak{\iota} }\frac{\mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{w}\mathbb{\lambda}}\;\;\right) \mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta }\right) \mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2} }d\mathfrak{w}\\ \leq\left[ \left( 1-\mathbb{\rho}\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\;\;\right) +\mathbb{\rho\zeta}\left( \mathbb{\tau}\right) \right] \int_{0}^{\mathfrak{\iota}}\frac{\mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\;\;\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.6)

By using the identity (2.2) in (2.6), we obtain (2.1).

Remark 1. If we choose $\mathbb{\rho} = \frac{\mathbb{\vartheta} }{\mathbb{\vartheta}+q}$ , $\mathbb{\theta} = \frac{1}{2}$ , $\mathfrak{\iota } = 2\breve{\varkappa}$ in Theorem 12, then

$\begin{array}{l} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\left( \mathbb{\vartheta}+q\right) }{\mathbb{\vartheta}\mathfrak{\varsigma }+q\mathbb{\tau}}\right) \int_{\frac{\mathbb{\lambda}}{1+\mathbb{\lambda\tau }\breve{\varkappa}}}^{\frac{\mathbb{\lambda}}{1-\mathbb{\lambda\tau} \breve{\varkappa}}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\int_{\frac{\mathbb{\lambda} }{1+\mathbb{\lambda\tau}\breve{\varkappa}}}^{\frac{\mathbb{\lambda} }{1-\mathbb{\lambda\tau}\breve{\varkappa}}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\left[ \frac{\mathbb{\vartheta\zeta}\left( \mathbb{\tau}\right) +q\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) }{\mathbb{\vartheta} +q}\right] \int_{\frac{\mathbb{\lambda}}{1+\mathbb{\lambda\tau} \breve{\varkappa}}}^{\frac{\mathbb{\lambda}}{1-\mathbb{\lambda\tau} \breve{\varkappa}}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.7)

Remark 2. If we choose $\mathbb{\rho} =$ $\mathbb{\theta} = \frac{1}{2}$ , $\mathfrak{\iota} = \mathbb{\tau}_{0} = \frac{\mathfrak{\varsigma}\mathbb{\tau} }{\mathbb{\tau}-\mathfrak{\varsigma}}$ in Theorem 12, then we get (1.8).

Remark 3. If we choose $\mathbb{\rho} =$ $\mathbb{\theta} = \frac{1}{2}$ , $\mathfrak{\iota} = \mathbb{\tau}_{0} = \frac{\mathfrak{\varsigma}\mathbb{\tau} }{\mathbb{\tau}-\mathfrak{\varsigma}}$ and $\mathbb{\vartheta}\left(\mathfrak{w}\right) = 1$ , $\mathfrak{w\in}\left[ \mathfrak{\varsigma }, \mathbb{\tau}\right]$ in Theorem 12, then we get (1.5).

Theorem 13. Let $\mathbb{\zeta}$ , $\mathbb{\lambda}$ and $\mathbb{\tau}_{0}$ be defined as in Theorem 12, $0 < \mathbb{\rho} < 1$ , $0 < \mathbb{\theta} < 1$ , $\mathbb{\rho}+\mathbb{\theta}\leq1$ and let $\mathbb{X}$ be defined on $\left[ 0, 1\right]$ as

$\begin{array}{l} \mathbb{X}\left( \mathfrak{\iota}\right) = \frac{\mathbb{\rho} \mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \\ \times\int_{0}^{\frac{\mathbb{\rho}\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) }}\left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota w}\mathbb{\lambda} }\right) +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota w}\mathbb{\lambda}}\right) \right] d\mathfrak{w}. \end{array}$

(2.8)

Then, $\mathbb{X}$ is harmonically convex on $\left(0, 1\right]$ andmonotonically increasing on $\left[ 0, 1\right]$ , and

$\begin{array}{l} \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho }\mathfrak{\varsigma}+\left( 1-\mathbb{\rho}\right) \mathbb{\tau}}\right) \leq\mathbb{X}\left( \mathfrak{\iota}\right) \leq\mathbb{X}\left( 1\right) = \frac{\mathbb{\rho}\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \label{C2}\\ \times\int_{0}^{\frac{\mathbb{\rho}\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) }}\left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{w}\mathbb{\lambda}}\right) +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{w}\mathbb{\lambda}}\right) \right] d\mathfrak{w}\\ \leq\left( 1-\mathbb{\rho}\right) \mathbb{\zeta}\left( \mathfrak{\varsigma }\right) +\mathbb{\rho\zeta}\left( \mathbb{\tau}\right) .\nonumber \end{array}$

Proof. Since $\mathbb{\zeta}$ is harmonically convex on $\left[ \mathfrak{\varsigma }, \mathbb{\tau}\right]$ this prove the harmonic convexity of $\mathbb{X}$ on $\left(0, 1\right]$ . By using the condition $\mathbb{\rho}+\mathbb{\theta }\leq1$ implies that $\mathbb{\tau}_{0} = \frac{\mathbb{\rho}\mathfrak{\varsigma }\mathbb{\tau}}{\left(1-\mathbb{\theta}\right) \left(\mathbb{\tau }-\mathfrak{\varsigma}\right) }$ . Since the mapping $\mathcal{W}:\left[ 0, \mathbb{\tau}_{0}\right] \rightarrow \mathbb{R}$ defined by

$\begin{equation} \mathcal{W}\left( \mathfrak{\iota}\right) = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}\mathfrak{\iota }\mathbb{\lambda}}\right) +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda }}{1+\left( 1-\mathbb{\theta}\right) \mathfrak{\iota}\mathbb{\lambda} }\right) \end{equation}$

(2.9)

has been proved to be monotonically increasing on $\left[ 0, \mathbb{\tau} _{0}\right]$ , thus the mapping $\mathbb{X}$ is also monotonically increasing on $\left[ 0, 1\right]$ .

Because X is monotonically increasing on $\left[ 0, 1\right]$ , it follows that the inequalities (2.8) can be deduced from these inequalities (2.5). The proof of the theorem was completed as a result of this.

The next theorem can be proved similarly:

Theorem 14. Let $\mathbb{\zeta}$ , $\mathbb{\lambda}$ , $\mathbb{\tau}_{0}$ , $\mathbb{\rho}$ and $\mathbb{\theta}$ be defined as in Theorem 13. Let $\mathbb{X}_{1}$ be defined on $\left[ 0, 1\right]$ as

$\begin{array}{l} \mathbb{X}_{1}\left( \mathfrak{\iota}\right) = \frac{\mathbb{\rho }\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \\ \times\int_{0}^{\frac{\mathbb{\rho}\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \left( \mathbb{\tau}-\mathfrak{\varsigma}\right) }}\left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma }\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma }\mathbb{\tau}-\mathbb{\theta}\left( \mathbb{\rho}\left( \mathbb{\tau }-\mathfrak{\varsigma}\right) -\mathfrak{w}\left( 1-\mathfrak{\iota}\right) \left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau}\right) \mathbb{\lambda}}\right) \right. \\ \left. +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau}+\left( 1-\mathbb{\theta}\right) \left( \mathbb{\rho}\left( \mathbb{\tau }-\mathfrak{\varsigma}\right) -\mathfrak{w}\left( 1-\mathfrak{\iota}\right) \left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau}\right) \mathbb{\lambda}}\right) \right] d\mathfrak{w}. \end{array}$

(2.10)

Then, $\mathbb{X}_{1}$ is harmonically convex monotonically increasing on $\left[ 0, 1\right]$ , and

$\begin{array}{l} \frac{\left( 1-\mathbb{\theta}\right) ^{2}\mathfrak{\varsigma}\mathbb{\tau} }{\mathbb{\rho\theta}\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \int_{\mathbb{\lambda}}^{\frac{\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}-\mathbb{\rho}\left( \mathbb{\tau}-\mathfrak{\varsigma }\right) }}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w} ^{2}}d\mathfrak{w}+\frac{\mathbb{\theta}\mathfrak{\varsigma}\mathbb{\tau} }{\mathbb{\rho}\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \int_{\mathfrak{\varsigma}}^{\mathbb{\lambda}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\mathbb{X}_{1}\left( \mathfrak{\iota}\right) \leq\mathbb{X}_{1}\left( 1\right) = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\left( 1-\mathbb{\theta}\right) \mathfrak{\varsigma}\mathbb{\tau} }{\left( \mathfrak{\varsigma}-\mathbb{\tau}\right) \mathbb{\rho}+\left( 1-\mathbb{\theta}\right) \mathbb{\tau}}\right) +\mathbb{\theta\zeta}\left( \mathfrak{\varsigma}\right) \\ \leq\left( 1-\mathbb{\rho}\right) \mathbb{\zeta}\left( \mathfrak{\varsigma }\right) +\mathbb{\rho\zeta}\left( \mathbb{\tau}\right) . \end{array}$

(2.11)

Remark 4. Taking $\mathbb{\rho} = \mathbb{\theta} = \frac{1}{2}$ in the inequality (2.8) reduces to

$\mathbb{S}\left( \mathfrak{\iota}\right) = \frac{\mathfrak{\varsigma }\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma} \mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\left( \mathfrak{\varsigma }+\mathbb{\tau}\right) }\right) \frac{d\mathfrak{w}}{\mathfrak{w}^{2}}.$

Remark 5. Taking $\mathbb{\rho} = \mathbb{\theta} = \frac{1}{2}$ in the inequality (2.10) reduces to

$\begin{array}{l} \mathbb{V}\left( \mathfrak{\iota}\right) = \frac{\mathfrak{\varsigma }\mathbb{\tau}}{2\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) } \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathbb{\tau}\mathfrak{w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) \right] d\mathfrak{w}. \end{array}$

(2.12)

Theorem 15. Let $\mathbb{\zeta}$ , $\mathbb{\rho}$ , $\mathbb{\theta}$ , $\mathbb{\lambda}$ , $\mathbb{\tau}_{0}$ be defined as in Theorem 13 and let $\mathbb{\vartheta}$ be defined as in Theorem 12. Let $\mathbb{Y}$ be a function defined on $\left[ 0, 1\right]$ by

$\begin{array}{l} \mathbb{Y}\left( \mathfrak{\iota}\right) = \int_{0}^{s}\left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda} }{1-\mathbb{\theta\lambda}\mathfrak{\iota w}}\right) \mathbb{\vartheta }\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda}\mathfrak{w}}\right) \right. \\ \left. +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathbb{\lambda}\mathfrak{\iota w}}\right) \mathbb{\vartheta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta }\right) \mathbb{\lambda}\mathfrak{w}}\right) \right] d\mathfrak{w} \end{array}$

(2.13)

for some $s\in\left[ 0, \mathbb{\tau}_{0}\right]$ . Then $\mathbb{Y}$ isharmonic convex and monotonically increasing on $\left(0, 1\right]$ and

(2.14)

Proof. Since $\mathbb{\zeta}$ is harmonic convex and $\mathbb{\vartheta}$ is nonnegative, we see that $\mathbb{Y}$ is harmonic convex on $\left(0, 1\right]$ . Let $\mathfrak{w}\in\left[ 0, s\right]$ , where $s\in\left[ 0, \mathbb{\tau}_{0}\right]$ , from Theorem 12 we get $\breve {h}\left(\mathfrak{\iota w}\right) = \left(1-\mathbb{\theta}\right) \mathbb{\zeta}\left(\frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda }\mathfrak{\iota w}}\right) +\mathbb{\theta\zeta}\left(\frac {\mathbb{\lambda}}{1+\left(1-\mathbb{\theta}\right) \mathbb{\lambda }\mathfrak{\iota w}}\right)$ is increasing for $\mathfrak{\iota}\in\left[ 0, 1\right]$ . Therefore the inequalities (2.14) are achieved immediately.

Theorem 16. Let $\mathbb{\zeta}$ , $\mathbb{\rho}$ , $\mathbb{\theta}$ , $\mathbb{\lambda}$ , $\mathbb{\tau}_{0}$ be defined as in Theorem 15 and let $\mathbb{\vartheta}$ be defined as in Theorem 12. Let $\mathbb{Y}_{1}$ be a function defined on $\left[ 0, 1\right]$ by

$\begin{array}{l} \mathbb{Y}_{1}\left( \mathfrak{\iota}\right) = \int_{0}^{s}\left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda} }{1-\mathbb{\theta}s+\mathbb{\theta}\mathfrak{w}\left( 1-\mathfrak{\iota }\right) \mathbb{\lambda}}\right) \mathbb{\vartheta}\left( \frac {\mathbb{\lambda}}{1-\mathbb{\theta}\left( s-\mathfrak{w}\right) \mathbb{\lambda}}\right) \right. \\ \left. +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) s-\left( 1-\mathbb{\theta}\right) \mathfrak{w} \left( 1-\mathfrak{\iota}\right) \mathbb{\lambda}}\right) \mathbb{\vartheta }\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \left( s-\mathfrak{w}\right) \mathbb{\lambda}}\right) \right] d\mathfrak{w} \end{array}$

(2.15)

for some $s\in\left[ 0, \mathbb{\tau}_{0}\right]$ . Then $\mathbb{Y}_{1}$ isharmonic convex $\left(0, 1\right]$ and monotonically increasing on $\left[ 0, 1\right]$ , and

$\begin{array}{l} \frac{1-\mathbb{\theta}}{\mathbb{\theta}}\int_{\frac{\mathbb{\lambda} }{1-\mathbb{\theta\lambda}s}}^{\mathbb{\lambda}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\frac{\mathbb{\theta}}{1-\mathbb{\theta}} \int_{\mathbb{\lambda}}^{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta }\right) \mathbb{\lambda}s}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} \\ \leq\mathbb{Y}\left( \mathfrak{\iota}\right) \leq\mathbb{Y}\left( 1\right) = \left[ \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda}}{1-\mathbb{\theta}s\mathbb{\lambda}}\right) +\left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda} }{1+\left( 1-\mathbb{\theta}\right) s\mathbb{\lambda}}\right) \right] \\ \times\int_{\frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda}s}}^{\frac {\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathbb{\lambda}s} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}\leq\left[ \left( 1-\mathbb{\rho}\right) \mathbb{\zeta }\left( \mathfrak{\varsigma}\right) +\mathbb{\rho\zeta}\left( \mathbb{\tau }\right) \right] \int_{\frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda}s} }^{\frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \mathbb{\lambda }s}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}. \end{array}$

(2.16)

Proof. Since $\mathbb{\zeta}$ is harmonic convex and $\mathbb{\vartheta}$ is nonnegative, we see that $\mathbb{Y}_{1}$ is harmonic convex on $\left(0, 1\right]$ . Next, for each $\mathfrak{w}\in\left[ 0, \mathfrak{\iota }\right]$ , where $\mathfrak{\iota}\in\left[ 0, \mathbb{\tau}_{0}\right]$ , it follows from Theorem 12 that $\breve{h}\left(\mathfrak{\iota }\right) = \left(1-\mathbb{\theta}\right) \mathbb{\zeta}\left(\frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda}\mathfrak{\iota}}\right) +\mathbb{\theta\zeta}\left(\frac{\mathbb{\lambda}}{1+\left(1-\mathbb{\theta}\right) \mathbb{\lambda}\mathfrak{\iota}}\right)$ and $k\left(\mathfrak{\iota}\right) = s-\left(1-\mathfrak{\iota}\right) \mathfrak{w}$ are increasing on $\left[ 0, \mathbb{\tau}_{0}\right]$ and $\left[ 0, 1\right]$ respectively. Hence

$\begin{array}{l} \breve{h}\left( k\left( \mathfrak{\iota}\right) \right) = \left( 1-\mathbb{\theta}\right) \mathbb{\zeta}\left( \frac{\mathbb{\lambda} }{1-\mathbb{\theta}s+\mathbb{\theta}\mathfrak{w}\left( 1-\mathfrak{\iota }\right) \mathbb{\lambda}}\right) \mathbb{\vartheta}\left( \frac {\mathbb{\lambda}}{1-\mathbb{\theta}\left( s-\mathfrak{w}\right) \mathbb{\lambda}}\right) \\ +\mathbb{\theta\zeta}\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) s-\left( 1-\mathbb{\theta}\right) \mathfrak{w} \left( 1-\mathfrak{\iota}\right) \mathbb{\lambda}}\right) \mathbb{\vartheta }\left( \frac{\mathbb{\lambda}}{1+\left( 1-\mathbb{\theta}\right) \left( s-\mathfrak{w}\right) \mathbb{\lambda}}\right) \end{array}$

is increasing on $\left[ 0, 1\right]$ . Using the identity $\mathbb{\vartheta }\left(\frac{\mathbb{\lambda}}{1-\mathbb{\theta\lambda}\mathfrak{\iota} }\right) = \mathbb{\vartheta}\left(\frac{\mathbb{\lambda}}{1+\left(1-\mathbb{\theta}\right) \mathbb{\lambda}\mathfrak{\iota}}\right)$ we see that $\mathbb{Y}\left(\mathfrak{\iota}\right)$ is increasing on $\left[ 0, 1\right]$ . Therefore the inequalities (2.16) follows from

$\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\rho }\mathfrak{\varsigma}+\left( 1-\mathbb{\rho}\right) \mathbb{\tau}}\right) \leq\mathcal{W}\left( k\left( \mathfrak{\iota}\right) \right) \leq\left( 1-\mathbb{\rho}\right) \mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\rho\zeta}\left( \mathbb{\tau}\right)$

and (2.16).

Remark 6. Choose $\mathbb{\rho} = \mathbb{\theta} = \frac{1}{2}$ , $s = \mathbb{\tau}_{0} = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau }-\mathfrak{\varsigma}}$ in Theorems 15 and 16. Then the inequalities (2.14) and (2.16) reduce to

(2.17)

where

$\mathbb{Y}\left( \mathfrak{\iota}\right) = \frac{\mathfrak{\varsigma }\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma} \mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \mathfrak{w}\left( \mathfrak{\varsigma }+\mathbb{\tau}\right) }\right) \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}$

and

$\begin{array}{l} \mathbb{Y}_{1}\left( \mathfrak{\iota}\right) = \frac{1}{2}\int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathbb{\tau}\mathfrak{w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) \mathbb{\vartheta}\left( \frac{2\mathfrak{\varsigma w}}{\mathfrak{\varsigma}+\mathfrak{w}}\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma w}}{\left( 1+\mathfrak{\iota}\right) \mathfrak{w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) \mathbb{\vartheta}\left( \frac{2\mathfrak{w} \mathbb{\tau}}{\mathfrak{w}+\mathbb{\tau}}\right) \right] d\mathfrak{w}. \end{array}$

(2.18)

Remark 7. The inequalities (2.17) provide weighted generalizations of Theorems 9 and 15.

In the coming results we provide weighted generalizations of Theorems 2–4 for harmonic convex functions by using Lemma 2.

Theorem 17. Let $\mathbb{\zeta}$ , $\mathbb{\vartheta}$ , $\mathbb{S}_{\mathbb{\vartheta}}$ be defined as above. Then

(i) The inequality

$\begin{array}{l} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq2\int_{\frac{4\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+3\mathbb{\tau}}}^{\frac {4\mathfrak{\varsigma}\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}} }\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{4\mathfrak{\varsigma }\mathbb{\tau}-\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \frac{d\mathfrak{w}}{\mathfrak{w}^{2}} \\ \leq\int_{0}^{1}\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota }\right) d\mathfrak{\iota}\leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta }\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\right] \end{array}$

(2.19)

holds.

(ii) If $\mathbb{\zeta}$ is differentiable on $\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ and $\mathbb{\vartheta}$ isbounded on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then theinequalities

$\begin{array}{l} 0\leq\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}-\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \\ \leq\left( 1-\mathfrak{\iota}\right) \left[ \left( \frac{\mathbb{\tau }-\mathfrak{\varsigma}}{\mathfrak{\varsigma}\mathbb{\tau}}\right) \left[ \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}\right] -\int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\right] \left\Vert \mathbb{\vartheta }\right\Vert _{\infty}, \end{array}$

(2.20)

hold for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ , where $\left\Vert\mathbb{\vartheta}\right\Vert _{\infty} = \underset{\mathfrak{w}\in\left[\mathfrak{\varsigma}, \mathbb{\tau}\right] }{\sup}\mathbb{\vartheta}\left(\mathfrak{w}\right)$ .

(iii) If $\mathbb{\zeta}$ is differentiable on $\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then, for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ , then

$\begin{array}{l} 0\leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}-\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \\ \leq\frac{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) \left( \mathbb{\tau}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathbb{\tau}\right) -\mathfrak{\varsigma}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{\varsigma }\right) \right) }{4\mathfrak{\varsigma}\mathbb{\tau}}\int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.21)

Proof. (i) Using techniques of integration and the hypothesis of $\mathbb{\vartheta}$ , we have the following identities:

$\begin{equation} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} = 4\int_{\mathfrak{\varsigma}}^{\frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}} \int_{0}^{\frac{1}{2}}\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) \frac {\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{\iota}d\mathfrak{w}, \end{equation}$

(2.22)

(2.23)

$\begin{array}{l} \int_{0}^{1}\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) d\mathfrak{\iota} \\ = \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\int_{0}^{\frac{1}{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}+2\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) \right] \\ \times\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{\iota}d\mathfrak{w}+\int_{\mathfrak{\varsigma}}^{\frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}} \int_{0}^{\frac{1}{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau }\right) \mathfrak{w}}\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{2\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w} \right) }{\mathfrak{w}^{2}}d\mathfrak{\iota}d\mathfrak{w} \end{array}$

(2.24)

and

$\begin{array}{l} \frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}+\int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\right] \\ = \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\int_{0}^{\frac{1}{2}}\left[ \mathbb{\zeta}\left( \mathfrak{w}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{\iota}d\mathfrak{w}\\ +\int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\int_{0}^{\frac{1}{2}}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{\mathfrak{\varsigma }+\mathbb{\tau}-\mathfrak{w}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{\iota}d\mathfrak{w}. \end{array}$

(2.25)

By using Lemma 2, we observe that the following inequalities hold for all $\mathfrak{\iota}\in\left[ 0, \frac{1}{2}\right]$ and $\mathfrak{w} \in\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ :

$\begin{equation} 4\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \leq2\left[ \mathbb{\zeta }\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}} {2\mathfrak{\varsigma}\mathbb{\tau}+\left( \mathfrak{\varsigma}+\mathbb{\tau }\right) \mathfrak{w}}\right) +\mathbb{\zeta}\left( \frac {4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{3\left( \mathfrak{\varsigma }+\mathbb{\tau}\right) \mathfrak{w}-2\mathfrak{\varsigma}\mathbb{\tau} }\right) \right] , \end{equation}$

(2.26)

$\begin{array}{l} 2\mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}+\left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) \\ \leq\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} \mathfrak{w}}{\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau }\right) \mathfrak{w}+2\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) , \end{array}$

(2.27)

$\begin{array}{l} 2\mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{3\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w} -2\mathfrak{\varsigma}\mathbb{\tau}}\right) \leq\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \\ +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) , \end{array}$

(2.28)

$\begin{array}{l} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}+2\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) \\ \leq\mathbb{\zeta}\left( \mathfrak{w}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) \end{array}$

(2.29)

and

$\begin{array}{l} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w} -\mathfrak{\varsigma}\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \\ +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \\ \leq\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{\mathfrak{\varsigma }+\mathbb{\tau}-\mathfrak{w}}\right) . \end{array}$

(2.30)

Multiplying the inequalities (2.26)–(2.30) by $\frac {\mathbb{\vartheta}\left(\mathfrak{w}\right) }{\mathfrak{w}^{2}}$ and integrating them over $\mathfrak{\iota}$ on $\left[ 0, \frac{1}{2}\right]$ , over $\mathfrak{w}$ on $\left[ \mathfrak{\varsigma}, \frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right]$ and using identities (2.22)–(2.25), we derive (2.19).

(ii) Since $\mathbb{\zeta}:\left[ \mathfrak{\varsigma }, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is harmonic convex on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ , hence $\breve{h}:\left[ \frac{1}{\mathbb{\tau}}, \frac{1}{\mathfrak{\varsigma }}\right] \rightarrow \mathbb{R}$ defined by $\breve{h}\left(\mathfrak{w}\right) = \mathbb{\zeta}\left(\frac{1}{\mathfrak{w}}\right)$ is convex on $\left[ \frac{1}{\mathbb{\tau} }, \frac{1}{\mathfrak{\varsigma}}\right]$ . Thus, by integration by parts, we get that following identity holds:

$\begin{array}{l} \int_{\frac{1}{\mathbb{\tau}}}^{\frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}}\left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \left[ \breve{h}^{^{\prime}}\left( \frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\mathfrak{w}\right) -\breve{h}^{^{\prime}}\left( \mathfrak{w}\right) \right] d\mathfrak{w} \\ = \left( \frac{\mathbb{\tau}-\mathfrak{\varsigma}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) \left[ \breve{h}\left( \frac{1}{\mathfrak{\varsigma }}\right) +\breve{h}\left( \frac{1}{\mathbb{\tau}}\right) \right] -\int_{\frac{1}{\mathbb{\tau}}}^{\frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}}\left[ \breve{h}\left( \frac {1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\breve{h}\left( \mathfrak{w}\right) \right] d\mathfrak{w}. \end{array}$

(2.31)

The equality (2.31) is equivalent to the equality:

$\begin{array}{l} \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\frac{1}{\mathfrak{w}^{2}}\left( \frac{1}{\mathfrak{w}}-\frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right) \left[ \frac{\mathbb{\zeta }^{^{\prime}}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\frac{1}{\mathfrak{w}}}\right) }{\left( \frac {1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\frac{1}{\mathfrak{w} }\right) ^{2}}-\mathfrak{w}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{w}\right) \right] d\mathfrak{w} \\ = \left( \frac{\mathbb{\tau}-\mathfrak{\varsigma}}{\mathfrak{\varsigma }\mathbb{\tau}}\right) \left[ \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2}\right] -\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta }\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.32)

Using substitution rules for integration and the hypothesis of $\mathbb{\vartheta}$ , we have the following identities:

$\begin{equation} \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} = \int_{\mathfrak{\varsigma}}^{\frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}} }\left[ \mathbb{\zeta}\left( \mathfrak{w}\right) +\mathbb{\zeta}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\frac {1}{\mathfrak{w}}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \end{equation}$

(2.33)

and

$\begin{array}{l} \mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) = \int _{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau }\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w} \right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.34)

Now, using the convexity of $\breve{h}\left(\mathfrak{w}\right) = \mathbb{\zeta}\left(\frac{1}{\mathfrak{w}}\right)$ on $\left[ \frac {1}{\mathbb{\tau}}, \frac{1}{\mathfrak{\varsigma}}\right]$ and the hypothesis of $\mathbb{\vartheta}$ , the following inequality holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ and $\mathfrak{w}\in\left[ \frac {1}{\mathbb{\tau}}, \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right]$ :

$\begin{array}{l} \left[ \breve{h}\left( \mathfrak{w}\right) -\breve{h}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac {\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w} }\right) \\ +\left[ \breve{h}\left( \frac{1}{\mathfrak{\varsigma}}+\frac{1} {\mathbb{\tau}}-\mathfrak{w}\right) -\breve{h}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) \\ \leq\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{w}-\frac {\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \breve{h}^{^{\prime}}\left( \mathfrak{w}\right) \mathbb{\vartheta }\left( \frac{1}{\mathfrak{w}}\right) \\ +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \breve{h}^{^{\prime}}\left( \frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\mathfrak{w}\right) \mathbb{\vartheta}\left( \frac {1}{\mathfrak{w}}\right) \\ = \left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \left[ \breve{h}^{^{\prime}}\left( \frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\mathfrak{w}\right) -\breve{h}^{^{\prime}}\left( \mathfrak{w}\right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w} }\right) \end{array}$

(2.35)

which is equivalent to

$\begin{array}{l} \left[ \mathbb{\zeta}\left( \frac{1}{\mathfrak{w}}\right) -\mathbb{\zeta }\left( \frac{1}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) }\right) \right] \mathbb{\vartheta}\left( \frac {1}{\mathfrak{w}}\right) \\ +\left[ \mathbb{\zeta}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma}} +\frac{1}{\mathbb{\tau}}-\mathfrak{w}}\right) -\mathbb{\zeta}\left( \frac {1}{\mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) }\right) \right] \mathbb{\vartheta}\left( \frac {1}{\mathfrak{w}}\right) \\ \leq\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \left[ \frac{\mathbb{\zeta}^{^{\prime}}\left( \frac{1}{\mathfrak{w}}\right) }{\mathfrak{w}^{2}}-\frac{\mathbb{\zeta}^{^{\prime}}\left( \frac{1}{\frac {1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}}\right) }{\left( \frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}} -\mathfrak{w}\right) ^{2}}\right] \mathbb{\vartheta}\left( \frac {1}{\mathfrak{w}}\right) . \end{array}$

(2.36)

Integrating the above inequalities over $\mathfrak{w}$ on $\left[ \frac {1}{\mathbb{\tau}}, \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right]$ , we get

$\begin{array}{l} \int_{\frac{1}{\mathbb{\tau}}}^{\frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}}\left[ \mathbb{\zeta}\left( \frac {1}{\mathfrak{w}}\right) -\mathbb{\zeta}\left( \frac{1}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) }\right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) d\mathfrak{w} \\ +\int_{\frac{1}{\mathbb{\tau}}}^{\frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}}\left[ \mathbb{\zeta}\left( \frac {1}{\frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w} }\right) -\mathbb{\zeta}\left( \frac{1}{\mathfrak{\iota}\left( \frac {1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) }\right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) d\mathfrak{w}\\ \leq\left( 1-\mathfrak{\iota}\right) \int_{\frac{1}{\mathbb{\tau}}} ^{\frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau }}}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}-\mathfrak{w}\right) \left[ \frac{\mathbb{\zeta}^{^{\prime} }\left( \frac{1}{\mathfrak{w}}\right) }{\mathfrak{w}^{2}}-\frac {\mathbb{\zeta}^{^{\prime}}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma} }+\frac{1}{\mathbb{\tau}}-\mathfrak{w}}\right) }{\left( \frac{1} {\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) ^{2} }\right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) d\mathfrak{w}. \end{array}$

(2.37)

After making use of suitable substitution, the inequality (2.37) takes the form:

$\begin{array}{l} \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{1}{\mathfrak{\iota}\frac{1}{\mathfrak{w}}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}\right) }\right) -\mathbb{\zeta}\left( \mathfrak{w}\right) \right] \mathbb{\vartheta}\left( \mathfrak{w}\right) d\mathfrak{w} \\ +\int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\frac{1}{\mathfrak{w}^{2}}\left[ \mathbb{\zeta}\left( \frac{1}{\mathfrak{\iota}\left( \frac{1} {\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\frac{1}{\mathfrak{w}}\right) +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) }\right) -\mathbb{\zeta}\left( \frac{1}{\frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\frac{1}{\mathfrak{w}}}\right) \right] \mathbb{\vartheta }\left( \mathfrak{w}\right) d\mathfrak{w}\\ \leq\left\Vert \mathbb{\vartheta}\right\Vert _{\infty}\left( 1-\mathfrak{\iota}\right) \int_{\mathfrak{\varsigma}}^{\frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}} \frac{1}{\mathfrak{w}^{2}}\left( \frac{1}{\mathfrak{w}}-\frac {\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \left[ \frac{\mathbb{\zeta}^{^{\prime}}\left( \frac{1}{\frac {1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau}}-\frac{1}{\mathfrak{w}} }\right) }{\left( \frac{1}{\mathfrak{\varsigma}}+\frac{1}{\mathbb{\tau} }-\frac{1}{\mathfrak{w}}\right) ^{2}}-\mathfrak{w}^{2}\mathbb{\zeta }^{^{\prime}}\left( \mathfrak{w}\right) \right] d\mathfrak{w}. \end{array}$

(2.38)

Inequality (2.20) follows from (2.31)–(2.34) and (2.38).

(iii) We use the fact that $\mathbb{\zeta }:\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right] \rightarrow \mathbb{R}$ is harmonic convex on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ , hence $\breve{h}:\left[ \frac{1}{\mathbb{\tau}}, \frac{1}{\mathfrak{\varsigma }}\right] \rightarrow \mathbb{R}$ defined by $\breve{h}\left(\mathfrak{w}\right) = \mathbb{\zeta}\left(\frac{1}{\mathfrak{w}}\right)$ is convex on $\left[ \frac{1}{\mathbb{\tau} }, \frac{1}{\mathfrak{\varsigma}}\right]$ . Thus

$\frac{\breve{h}\left( \frac{1}{\mathbb{\tau}}\right) -\breve{h}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) }{2}\leq\frac{\mathfrak{\varsigma}-\mathbb{\tau}} {4\mathfrak{\varsigma}\mathbb{\tau}}\breve{h}^{^{\prime}}\left( \frac {1}{\mathbb{\tau}}\right)$

and

$\frac{\breve{h}\left( \frac{1}{\mathfrak{\varsigma}}\right) -\breve {h}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) }{2}\leq\frac{\mathbb{\tau}-\mathfrak{\varsigma} }{4\mathfrak{\varsigma}\mathbb{\tau}}\breve{h}^{^{\prime}}\left( \frac {1}{\mathfrak{\varsigma}}\right).$

Adding the above inequalities

$\begin{equation} \frac{\breve{h}\left( \frac{1}{\mathfrak{\varsigma}}\right) +\breve {h}\left( \frac{1}{\mathbb{\tau}}\right) }{2}-\breve{h}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \leq\frac{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) \left( \breve{h}^{^{\prime}}\left( \frac{1}{\mathfrak{\varsigma}}\right) -\breve {h}^{^{\prime}}\left( \frac{1}{\mathbb{\tau}}\right) \right) } {4\mathfrak{\varsigma}\mathbb{\tau}}. \end{equation}$

(2.39)

The inequality (2.39) becomes

$\begin{equation} \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}-\mathbb{\zeta}\left( \frac {2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) \leq\frac{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) \left( \mathbb{\tau}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathbb{\tau}\right) -\mathfrak{\varsigma}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{\varsigma }\right) \right) }{4\mathfrak{\varsigma}\mathbb{\tau}}. \end{equation}$

(2.40)

Multiplying (2.40) both sides by $\frac{\mathbb{\vartheta}\left(\mathfrak{w}\right) }{\mathfrak{w}^{2}}$ and integrating over $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ , we get

$\begin{array}{l} \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}-\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\frac{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) \left( \mathbb{\tau}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathbb{\tau}\right) -\mathfrak{\varsigma}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{\varsigma }\right) \right) }{4\mathfrak{\varsigma}\mathbb{\tau}}\int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.41)

From (2.17) and (2.41) we get (2.21).

Corollary 1. Suppose that the assumption of Theorem 17 are satisfiedand $\mathbb{\vartheta}\left(\mathfrak{w}\right) = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}$ , $\mathfrak{w}\in\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then

(i) The inequalities

$\begin{array}{l} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \leq2\int_{\frac {4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+3\mathbb{\tau}} }^{\frac{4\mathfrak{\varsigma}\mathbb{\tau}}{3\mathfrak{\varsigma }+\mathbb{\tau}}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}\leq\int_{0}^{1}\mathbb{S}\left( \mathfrak{\iota}\right) d\mathfrak{\iota} \\ \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\frac {\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}} \int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\right] \end{array}$

(2.42)

holds.

(ii) The inequalities

$\begin{array}{l} 0\leq\frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau} -\mathfrak{\varsigma}}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}} \frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}-\mathbb{S}\left( \mathfrak{\iota}\right) \\ \leq\left( 1-\mathfrak{\iota}\right) \left[ \frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2}-\frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma }}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\right] \end{array}$

(2.43)

hold for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

(iii) The inequalities

$\begin{equation} 0\leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}-\mathbb{S}\left( \mathfrak{\iota}\right) \leq\frac{\left( \mathbb{\tau}-\mathfrak{\varsigma}\right) \left( \mathbb{\tau}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathbb{\tau}\right) -\mathfrak{\varsigma}^{2}\mathbb{\zeta}^{^{\prime}}\left( \mathfrak{\varsigma }\right) \right) }{4\mathfrak{\varsigma}\mathbb{\tau}} \end{equation}$

(2.44)

are valid for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

In the following theorems, we discuss inequalities for the functions $\mathbb{S}$ , $\mathbb{S}_{\mathbb{\vartheta}}$ , $\mathcal{\breve{G}}_{1}$ , $\mathbb{T}$ and $\mathbb{T}_{\mathbb{\vartheta}}$ as considered above:

Theorem 18. Let $\mathbb{\zeta}$ , $\mathbb{\vartheta}$ , $\mathcal{\breve{G}}_{1}$ , $\mathbb{S}_{\mathbb{\vartheta}}$ be defined as above. Then

(i) The inequality

$\begin{equation} \mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \leq \mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right) \int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \end{equation}$

(2.45)

holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

(ii) The inequalities

$\begin{array}{l} 2\int_{\frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}}^{\frac{4\mathfrak{\varsigma}\mathbb{\tau}} {3\mathfrak{\varsigma}+\mathbb{\tau}}}\mathbb{\zeta}\left( \mathfrak{w} \right) \mathbb{\vartheta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{4\mathfrak{\varsigma}\mathbb{\tau}-\left( \mathfrak{\varsigma }+\mathbb{\tau}\right) \mathfrak{w}}\right) \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma }\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}\right) \right] \int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}\\ \leq\frac{\mathbb{\tau}-\mathfrak{\varsigma}}{\mathfrak{\varsigma} \mathbb{\tau}}\int_{0}^{1}\mathbb{\lambda}_{1}\left( \mathfrak{\iota}\right) \mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}+\mathfrak{\iota}\mathbb{\tau} }\right) d\mathfrak{\iota}\\ \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\frac {\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}\right] \int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} \end{array}$

(2.46)

hold.

(iii) If $\mathbb{\zeta}$ is differentiable on $\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ and $\mathbb{\vartheta}$ isbounded on $\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then, forall $\mathfrak{\iota}\in\left[ 0, 1\right]$ , then

$\begin{equation} 0\leq\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) -\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right) \int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\left( \frac{\mathbb{\tau} -\mathfrak{\varsigma}}{\mathfrak{\varsigma}\mathbb{\tau}}\right) \left[ \mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right) -\mathbb{S}\left( \mathfrak{\iota}\right) \right] \left\Vert \mathbb{\vartheta}\right\Vert _{\infty}, \end{equation}$

(2.47)

where $\left\Vert \mathbb{\vartheta}\right\Vert _{\infty} = \underset{\mathfrak{w}\in\left[ \mathfrak{\varsigma}, \mathbb{\tau}\right]}{\sup}\mathbb{\vartheta}\left(\mathfrak{w}\right)$ .

Proof. (i) Using integration by substitution and the assumptions on $\mathbb{\vartheta}$ , we have that the following identity holds on $\left[ 0, 1\right]$ :

$\begin{array}{l} \mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right) \int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} = \int _{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \right] \frac {\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}. \end{array}$

(2.48)

By Lemma 2, the following inequality holds for all $\mathfrak{w} \in\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ with

$\begin{align*} \mathfrak{w}_{1} & = \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w} }, \mathfrak{w}_{2} = \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau }\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}},\\ \breve{\varkappa}_{1} & = \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\;{\rm{and }}\;\breve{\varkappa} _{2} = \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathfrak{\varsigma\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma} +\mathbb{\tau}\right) }{:} \end{align*}$

$\begin{array}{l} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w} }\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{2\mathfrak{\iota}\left( \mathfrak{\varsigma w}+\mathbb{\tau }\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) +\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \\ \leq\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) . \end{array}$

(2.49)

Multiplying both sides of (2.49) with $\frac{\mathbb{\vartheta}\left(\mathfrak{w}\right) }{\mathfrak{w}^{2}}$ , integrating over $\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ and using (2.34) and (2.49), we obtain (2.45).

(ii) We can observe that

$\begin{array}{l} \frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma }\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}\right) \right] \int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} \\ = \left[ \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau} }{3\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+3\mathbb{\tau} }\right) \right] \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}}\frac{\mathbb{\vartheta }\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.50)

By using harmonic symmetric assumption on $\mathbb{\vartheta}$ , we get

(2.51)

We can also see that the following identity holds:

$\begin{array}{l} \frac{\mathbb{\tau}-\mathfrak{\varsigma}}{\mathfrak{\varsigma}\mathbb{\tau} }\int_{0}^{1}\mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right) \mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}+\mathfrak{\iota}\mathbb{\tau} }\right) d\mathfrak{\iota} = \frac{\mathbb{\tau}-\mathfrak{\varsigma} }{\mathfrak{\varsigma}\mathbb{\tau}} \\ \times\left[ \int_{\frac{1}{2}}^{1}\mathbb{\zeta}\left( \frac{1} {2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}+\mathfrak{\iota}\mathbb{\tau}}\right) d\mathfrak{\iota }\right. \\ +\int_{0}^{\frac{1}{2}}\mathbb{\zeta}\left( \frac{1}{2\mathfrak{\varsigma \iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma }+\mathbb{\tau}\right) }\right) \mathbb{\vartheta}\left( \frac {\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}+\mathfrak{\iota\varsigma}}\right) d\mathfrak{\iota}\\ +\int_{0}^{\frac{1}{2}}\mathbb{\zeta}\left( \frac{1}{2\mathbb{\tau }\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}+\mathfrak{\iota\varsigma}}\right) d\mathfrak{\iota}\\ \left. +\int_{\frac{1}{2}}^{1}\mathbb{\zeta}\left( \frac{1}{2\mathbb{\tau }\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}+\mathfrak{\iota}\mathbb{\tau}}\right) d\mathfrak{\iota }\right] = \frac{1}{2}\int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}}\left[ \mathbb{\zeta }\left( \frac{2\mathfrak{w\varsigma}}{\mathfrak{\varsigma}+\mathfrak{w} }\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{2\mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w} -\mathfrak{\varsigma}\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathbb{\tau}\mathfrak{w}}{\mathbb{\tau}+\mathfrak{w}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathfrak{\varsigma w}+2\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.52)

Finally we also have

$\begin{array}{l} \frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\frac {\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}\right] \int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} \\ = \left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2}\right] \int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}}\frac{\mathbb{\vartheta }\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}. \end{array}$

(2.53)

By Lemma 2, the following inequalities hold for all $\mathfrak{w} \in\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ :

The inequality

$\begin{equation} \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}+\left( \mathfrak{\varsigma} +\mathbb{\tau}\right) \mathfrak{w}}\right) +\mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{3\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}-2\mathfrak{\varsigma }\mathbb{\tau}}\right) \leq\mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma }\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}\right) \end{equation}$

(2.54)

holds with the choices of $\mathfrak{w}_{1} = \frac{4\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma}\mathbb{\tau}+\left(\mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}$ , $\mathfrak{w} _{2} = \frac{4\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{3\left(\mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}-2\mathfrak{\varsigma }\mathbb{\tau}}$ , $\breve{\varkappa}_{1} = \frac{4\mathfrak{\varsigma }\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}$ and $\breve{\varkappa }_{2} = \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}$ .

The inequality

$\begin{equation} \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}} {3\mathfrak{\varsigma}+\mathbb{\tau}}\right) \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma w} }{\mathfrak{\varsigma}+\mathfrak{w}}\right) \right] \end{equation}$

(2.55)

holds with the choices of $\mathfrak{w}_{1} = \mathfrak{w}_{2} = \frac {4\mathfrak{\varsigma}\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}$ , $\breve{\varkappa}_{1} = \frac{2\mathfrak{\varsigma w}}{\mathfrak{\varsigma }+\mathfrak{w}}$ , $\breve{\varkappa}_{2} = \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma w}+\mathbb{\tau} \mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}}$ .

The inequality

$\begin{equation} \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}} {\mathfrak{\varsigma}+3\mathbb{\tau}}\right) \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathfrak{\varsigma w}+2\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathbb{\tau }\mathfrak{w}}{\mathbb{\tau}+\mathfrak{w}}\right) \right] \end{equation}$

(2.56)

holds with the choices of $\mathfrak{w}_{1} = \mathfrak{w}_{2} = \frac {4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+3\mathbb{\tau}}$ , $\breve{\varkappa}_{1} = \frac{2\mathbb{\tau}\mathfrak{w}}{\mathbb{\tau }+\mathfrak{w}}$ , $\breve{\varkappa}_{2} = \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{\mathfrak{\varsigma w}+2\mathbb{\tau} \mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}}$ .

The inequality

$\begin{equation} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma w} }{\mathfrak{\varsigma}+\mathfrak{w}}\right) \leq\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \frac{2\mathbb{\tau }\mathfrak{\varsigma}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right) \end{equation}$

(2.57)

holds with the choices of $\mathfrak{w}_{1} = \frac{2\mathfrak{\varsigma w} }{\mathfrak{\varsigma}+\mathfrak{w}}$ , $\mathfrak{w}_{2} = \frac {2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{2\mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}}$ , $\breve{\varkappa}_{1} = \mathfrak{\varsigma}$ , $\breve{\varkappa}_{2} = \frac{2\mathbb{\tau}\mathfrak{\varsigma}}{\mathfrak{\varsigma}+\mathbb{\tau} }$ .

The inequality

$\begin{equation} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathfrak{\varsigma w}+2\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac{2\mathbb{\tau }\mathfrak{w}}{\mathbb{\tau}+\mathfrak{w}}\right) \leq\mathbb{\zeta}\left( \frac{2\mathbb{\tau}\mathfrak{\varsigma}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) \end{equation}$

(2.58)

holds with the choices of $\mathfrak{w}_{1} = \frac{2\mathbb{\tau}\mathfrak{w} }{\mathbb{\tau}+\mathfrak{w}}$ , $\mathfrak{w}_{2} = \frac{2\mathfrak{\varsigma }\mathbb{\tau}\mathfrak{w}}{\mathfrak{\varsigma w}+2\mathbb{\tau} \mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}}$ , $\breve{\varkappa}_{1} = \frac{2\mathbb{\tau}\mathfrak{\varsigma}}{\mathfrak{\varsigma}+\mathbb{\tau} }$ , $\breve{\varkappa}_{2} = \mathbb{\tau}$ .

Multiplying (2.54)–(2.58) by $\mathbb{\vartheta}\left(\mathfrak{w}\right)$ , integrating them over $\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma }\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ and using (2.50)–(2.53), we get (2.46).

(iii) By integration by parts, we get

$\begin{array}{l} \mathfrak{\iota}\int_{\frac{1}{\mathbb{\tau}}}^{\frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}}\left[ \left( \mathfrak{w}-\frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) \breve{h}^{^{\prime}}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right. \\ \left. +\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \breve{h}^{^{\prime }}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma}}+\frac {1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) \right) \right] d\mathfrak{w}\\ = \mathfrak{\iota}\int_{\frac{1}{\mathbb{\tau}}}^{\frac{1}{\mathfrak{\varsigma }}}\left( \mathfrak{w}-\frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \breve{h}^{^{\prime}}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac {\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \right) d\mathfrak{w}\\ = \frac{\mathbb{\tau}-\mathfrak{\varsigma}}{2\mathfrak{\varsigma}\mathbb{\tau} }\left[ \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) +\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \right] \\ -\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{1}{\mathfrak{w}^{2} }\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) d\mathfrak{w} = \left( \frac{\mathbb{\tau}-\mathfrak{\varsigma}} {\mathfrak{\varsigma}\mathbb{\tau}}\right) \left[ \mathcal{\breve{G}} _{1}\left( \mathfrak{\iota}\right) -\mathbb{S}\left( \mathfrak{\iota }\right) \right] . \end{array}$

(2.59)

Using the convexity of $\breve{h}$ and the hypothesis of $\mathbb{\vartheta}$ , the inequality holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ and $\mathfrak{w}\in\left[ \frac{1}{\mathbb{\tau}}, \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right]$ :

$\begin{array}{l} \left[ \breve{h}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) \right) -\breve{h}\left( \frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) \\ +\left[ \breve{h}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma }}+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota }\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) -\breve{h}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau} }\right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) \\ \leq\mathfrak{\iota}\left( \mathfrak{w}-\frac{\mathfrak{\varsigma }+\mathbb{\tau}}{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \breve {h}^{^{\prime}}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}}{2\mathfrak{\varsigma }\mathbb{\tau}}\right) \right) \mathbb{\vartheta}\left( \frac {1}{\mathfrak{w}}\right) \\ +\mathfrak{\iota}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \breve{h} ^{^{\prime}}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma} }+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota }\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \mathbb{\vartheta }\left( \frac{1}{\mathfrak{w}}\right) \\ = \mathfrak{\iota}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \left[ \breve {h}^{^{\prime}}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma} }+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota }\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right. \\ \left. -\breve{h}^{^{\prime}}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right] \mathbb{\vartheta}\left( \frac{1}{\mathfrak{w}}\right) \\ \leq\mathfrak{\iota}\left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}-\mathfrak{w}\right) \left[ \breve {h}^{^{\prime}}\left( \mathfrak{\iota}\left( \frac{1}{\mathfrak{\varsigma} }+\frac{1}{\mathbb{\tau}}-\mathfrak{w}\right) +\left( 1-\mathfrak{\iota }\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau}} {2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right. \\ \left. -\breve{h}^{^{\prime}}\left( \mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \left( \frac{\mathfrak{\varsigma}+\mathbb{\tau} }{2\mathfrak{\varsigma}\mathbb{\tau}}\right) \right) \right] \left\Vert \mathbb{\vartheta}\right\Vert _{\infty}. \end{array}$

(2.60)

Integrating (2.60), using (2.59) and (2.17), we get (2.47).

Corollary 2. According to the assumptions of Theorem 18 with $\mathbb{\vartheta}\left(\mathfrak{w}\right) = \frac{\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma}}$ , $\mathfrak{w}\in\left[\mathfrak{\varsigma}, \mathbb{\tau}\right]$ , then

(i) The inequality

$\mathbb{S}\left( \mathfrak{\iota}\right) \leq\mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right)$

holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

(ii) The inequalities

$\begin{array}{l} \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathbb{\tau}-\mathfrak{\varsigma} }\int_{\frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}}^{\frac{4\mathfrak{\varsigma}\mathbb{\tau}} {3\mathfrak{\varsigma}+\mathbb{\tau}}}\mathbb{\zeta}\left( \mathfrak{w} \right) \mathbb{\vartheta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{4\mathfrak{\varsigma}\mathbb{\tau}-\left( \mathfrak{\varsigma }+\mathbb{\tau}\right) \mathfrak{w}}\right) \frac{d\mathfrak{w} }{\mathfrak{w}^{2}} \\ \leq\frac{1}{2}\left[ \mathbb{\zeta}\left( \frac{4\mathfrak{\varsigma }\mathbb{\tau}}{3\mathfrak{\varsigma}+\mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{4\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+3\mathbb{\tau}}\right) \right] \leq\frac{\mathbb{\tau}-\mathfrak{\varsigma }}{\mathfrak{\varsigma}\mathbb{\tau}}\int_{0}^{1}\mathcal{\breve{G}} _{1}\left( \mathfrak{\iota}\right) \\ \times\mathbb{\vartheta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau} }{\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}+\mathfrak{\iota }\mathbb{\tau}}\right) d\mathfrak{\iota}\leq\frac{1}{2}\left[ \mathbb{\zeta }\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma }+\mathbb{\tau}}\right) +\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma }\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) }{2}\right] \end{array}$

(2.61)

hold.

(iii) The inequality

$\begin{equation} 0\leq\mathbb{S}\left( \mathfrak{\iota}\right) -\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}}{\mathfrak{\varsigma}+\mathbb{\tau} }\right) d\mathfrak{w}\leq\left( \frac{\mathbb{\tau}-\mathfrak{\varsigma} }{\mathfrak{\varsigma}\mathbb{\tau}}\right) \left[ \mathcal{\breve{G}} _{1}\left( \mathfrak{\iota}\right) -\mathbb{S}\left( \mathfrak{\iota }\right) \right] \end{equation}$

(2.62)

holds for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

Theorem 19. Let $\mathbb{\zeta}$ , $\mathbb{\vartheta}$ , $\mathcal{\breve{G}}_{1}$ , $\mathbb{S}_{\mathbb{\vartheta}}$ , $\mathbb{T}_{\mathbb{\vartheta}}$ bedefined as above. Then

(i) $\mathbb{T}_{\mathbb{\vartheta}}$ is harmonic convex on $\left(0, 1\right]$ .

(ii) The inequalities

$\begin{array}{l} \mathcal{\breve{G}}_{1}\left( \mathfrak{\iota}\right) \int _{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\leq\mathbb{T} _{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \\ \leq\left( 1-\mathfrak{\iota}\right) \int_{\mathfrak{\varsigma} }^{\mathbb{\tau}}\frac{\mathbb{\zeta}\left( \mathfrak{w}\right) \mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}+\mathfrak{\iota}\cdot\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau}\right) } {2}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w}\\ \leq\frac{\mathbb{\zeta}\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta }\left( \mathbb{\tau}\right) }{2}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau} }\frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}, \end{array}$

(2.63)

$\begin{equation} \mathbb{S}_{\mathbb{\vartheta}}\left( 1-\mathfrak{\iota}\right) \leq\mathbb{T}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \end{equation}$

(2.64)

and

$\begin{equation} \frac{\mathbb{S}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) +\mathbb{S}_{\mathbb{\vartheta}}\left( 1-\mathfrak{\iota}\right) }{2} \leq\mathbb{T}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) \end{equation}$

(2.65)

hold for all $\mathfrak{\iota}\in\left[ 0, 1\right]$ .

(iii) The following bound is true:

$\begin{equation} \underset{\mathfrak{\iota}\in\left[ 0,1\right] }{\sup}\mathbb{T} _{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) = \frac{\mathbb{\zeta }\left( \mathfrak{\varsigma}\right) +\mathbb{\zeta}\left( \mathbb{\tau }\right) }{2}\int_{\mathfrak{\varsigma}}^{\mathbb{\tau}}\frac {\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w}. \end{equation}$

(2.66)

Proof. (i) Since $\mathbb{\zeta}$ is harmonic convex and $\mathbb{\vartheta}$ is nonnegative, we see that $\mathbb{T}_{\mathbb{\vartheta}}$ is harmonic convex on $\left(0, 1\right]$ .

(ii) We observe that the following identity holds on $\left[ 0, 1\right]$ :

$\begin{array}{l} \mathbb{T}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) = \frac{1} {2}\int_{\mathfrak{\varsigma}}^{\frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}}\left[ \mathbb{\zeta}\left( \frac{\mathbb{\tau}\mathfrak{w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota }\right) \mathbb{\tau}}\right) +\mathbb{\zeta}\left( \frac {\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}}{\mathfrak{\varsigma w\iota }+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma}\mathbb{\tau}\right) }\right) \right. \\ \left. +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathbb{\tau}\mathfrak{w\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}\right) }\right) \right] \mathbb{\vartheta}\left( \mathfrak{w}\right) d\mathfrak{w}. \end{array}$

(2.67)

By Lemma 2, the following inequalities hold for all $\mathfrak{w} \in\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ :

$\begin{array}{l} 2\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \\ \leq\mathbb{\zeta}\left( \frac{\mathbb{\tau}\mathfrak{w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\right) +\mathbb{\zeta }\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w}} {\mathfrak{\varsigma w\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}\right) }\right) \end{array}$

(2.68)

with

$\begin{align*} \mathfrak{w}_{1} & = \mathfrak{w}_{2} = \frac{2\mathfrak{\varsigma} \mathbb{\tau}}{2\mathfrak{\varsigma\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }{, }\breve{\varkappa }_{1} = \frac{\mathbb{\tau}\mathfrak{w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}\\ {\rm{and }}\;\breve{\varkappa}_{2} & = \frac{\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{\mathfrak{\varsigma w\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}\right) }. \end{align*}$

$\begin{array}{l} 2\mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}} {2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }\right) \\ \leq\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{\mathbb{\tau}\mathfrak{w\iota}+\left( 1-\mathfrak{\iota}\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w}-\mathfrak{\varsigma }\mathbb{\tau}\right) }\right) \end{array}$

(2.69)

with

$\begin{align*} \mathfrak{w}_{1} & = \mathfrak{w}_{2} = \frac{2\mathfrak{\varsigma} \mathbb{\tau}}{2\mathbb{\tau}\mathfrak{\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma}+\mathbb{\tau}\right) }{, } \breve{\varkappa}_{1} = \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\\ {\rm{and }}\;\breve{\varkappa}_{2} & = \frac{\mathfrak{\varsigma}\mathbb{\tau }\mathfrak{w}}{\mathbb{\tau}\mathfrak{w\iota}+\left( 1-\mathfrak{\iota }\right) \left( \mathfrak{\varsigma w}+\mathbb{\tau}\mathfrak{w} -\mathfrak{\varsigma}\mathbb{\tau}\right) }. \end{align*}$

Multiplying the inequalities (2.68) and (2.69) by $\mathbb{\vartheta }\left(\mathfrak{w}\right)$ , integrating them over $\mathfrak{w}$ on $\left[ \mathfrak{\varsigma}, \frac{2\mathfrak{\varsigma}\mathbb{\tau} }{\mathfrak{\varsigma}+\mathbb{\tau}}\right]$ and using identities (2.48) and (2.67), we derive the first inequality of (2.63). Using the harmonic convexity of $\mathbb{\zeta}$ and the inequality (2.17), the last part of (2.63) holds. Using again the harmonic convexity of $\mathbb{\zeta}$ , we get

$\begin{array}{l} \mathbb{S}_{\mathbb{\vartheta}}\left( 1-\mathfrak{\iota}\right) = { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}\left( \frac{2\mathfrak{\varsigma}\mathbb{\tau}\mathfrak{w} }{2\mathfrak{\varsigma}\mathbb{\tau}\left( 1-\mathfrak{\iota}\right) +\mathfrak{\iota}\left( \mathfrak{\varsigma}+\mathbb{\tau}\right) \mathfrak{w}}\right) \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2}}d\mathfrak{w} \\ = { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \mathbb{\zeta}\left( \frac{1}{\frac{1}{2}\left( \frac{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}} {\mathfrak{\varsigma w}}\right) +\frac{1}{2}\left( \frac{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathbb{\tau}}{\mathbb{\tau} \mathfrak{w}}\right) }\right) \frac{\mathbb{\vartheta}\left( \mathfrak{w} \right) }{\mathfrak{w}^{2}}d\mathfrak{w}\\ \leq\frac{1}{2} { \int\limits_{\mathfrak{\varsigma}}^{\mathbb{\tau}}} \left[ \mathbb{\zeta}\left( \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) +\mathbb{\zeta}\left( \frac{\mathfrak{\varsigma w}}{\mathfrak{\iota w}+\left( 1-\mathfrak{\iota}\right) \mathfrak{\varsigma}}\right) \right] \frac{\mathbb{\vartheta}\left( \mathfrak{w}\right) }{\mathfrak{w}^{2} }d\mathfrak{w} = \mathbb{T}_{\mathbb{\vartheta}}\left( \mathfrak{\iota}\right) . \end{array}$

(2.70)

From (2.45), (2.63) and 2.70), we get (2.65).

(iii) (2.66) holds due to the inequality (2.63).

3. Conclusions

The subject of mathematical inequalities using convex functions has been seen to be an emerging topic during the past more than three decades. The researchers are trying to find new generalizations of convex functions and as a result new results are being adding to the theory of inequalities. In the current research we have used harmonic convex functions to generalize a number of results that hold for convex functions. In order to get the novel results in this study, we defined some new mappings over the interval $[0, 1]$ . We have discussed some interesting properties of these mappings and obtained new refinements of the Hermite-hadamard and Fejér type inequalities already proven for harmonic convex functions. We believe that the results of this paper could be a source of inspiration for mathematicians working in this field and young researchers thinking to start their career in this fascinating field of mathematics.

Acknowledgments

The author is very thankful to all the anonymous referees for their very useful and constructive comments in order to present the paper in the present form. This work is supported by the Deanship of Scientific Research, King Faisal University under the Nasher Track 2021 (Research Project Number NA000177) which has been converted to Ambitious Researcher Track (Research Project Number GRANT931).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

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