Graphs with mixed metric dimension three and related algorithms

Dalal Awadh Alrowaili; Uzma Ahmad; Saira Hameeed; Muhammad Javaid; Dalal Awadh Alrowaili; Uzma Ahmad; Saira Hameeed; Muhammad Javaid

doi:10.3934/math.2023854

AIMS Mathematics

2023, Volume 8, Issue 7: 16708-16723. doi: 10.3934/math.2023854

Previous Article Next Article

Research article Special Issues

Graphs with mixed metric dimension three and related algorithms

1.
Mathematics Department, College of Science, Jouf University, P.O. Box: 2014, Sakaka, Saudi Arabia
2.
Department of Mathematics, University of the Punjab, Lahore, Pakistan
3.
Department of Mathematics, School of Science, University of Management and Technology, Lahore 54770, Pakistan

Received: 15 November 2022 Revised: 07 April 2023 Accepted: 20 April 2023 Published: 12 May 2023
MSC : 05C12, 05C75

Let $G = (V, E)$ be a simple connected graph. A vertex $x\in V(G)$ resolves the elements $u, v\in E(G)\cup V(G)$ if $d_G(x, u)\neq d_G(x, v)$ . A subset $S\subseteq V(G)$ is a mixed metric resolving set for $G$ if every two elements of $G$ are resolved by some vertex of $S$ . A set of smallest cardinality of mixed metric generator for $G$ is called the mixed metric dimension. In this paper trees and unicyclic graphs having mixed dimension three are classified. The main aim is to investigate the structure of a simple connected graph having mixed dimension three with respect to the order of graph, maximum degree of basis elements and distance partite sets of basis elements. In particular to find necessary and sufficient conditions for a graph to have mixed metric dimension 3. Moreover three separate algorithms are developed for trees, unicyclic graphs and in general for simple connected graph $J_{n}\ncong P_{n}$ with $n\geq 3$ to determine "whether these graphs have mixed dimension three or not?". If these graphs have mixed dimension three, then these algorithms provide a mixed basis of an input graph.

Keywords:

Citation: Dalal Awadh Alrowaili, Uzma Ahmad, Saira Hameeed, Muhammad Javaid. Graphs with mixed metric dimension three and related algorithms[J]. AIMS Mathematics, 2023, 8(7): 16708-16723. doi: 10.3934/math.2023854

Related Papers:

[1]	Miguel Vivas-Cortez, Muhammad Aamir Ali, Artion Kashuri, Hüseyin Budak . Generalizations of fractional Hermite-Hadamard-Mercer like inequalities for convex functions. AIMS Mathematics, 2021, 6(9): 9397-9421. doi: 10.3934/math.2021546
[2]	Saad Ihsan Butt, Artion Kashuri, Muhammad Umar, Adnan Aslam, Wei Gao . Hermite-Jensen-Mercer type inequalities via Ψ-Riemann-Liouville k-fractional integrals. AIMS Mathematics, 2020, 5(5): 5193-5220. doi: 10.3934/math.2020334
[3]	Miguel Vivas-Cortez, Muhammad Uzair Awan, Muhammad Zakria Javed, Artion Kashuri, Muhammad Aslam Noor, Khalida Inayat Noor . Some new generalized $\kappa$ –fractional Hermite–Hadamard–Mercer type integral inequalities and their applications. AIMS Mathematics, 2022, 7(2): 3203-3220. doi: 10.3934/math.2022177
[4]	Jia-Bao Liu, Saad Ihsan Butt, Jamshed Nasir, Adnan Aslam, Asfand Fahad, Jarunee Soontharanon . Jensen-Mercer variant of Hermite-Hadamard type inequalities via Atangana-Baleanu fractional operator. AIMS Mathematics, 2022, 7(2): 2123-2141. doi: 10.3934/math.2022121
[5]	Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710
[6]	Yamin Sayyari, Mana Donganont, Mehdi Dehghanian, Morteza Afshar Jahanshahi . Strongly convex functions and extensions of related inequalities with applications to entropy. AIMS Mathematics, 2024, 9(5): 10997-11006. doi: 10.3934/math.2024538
[7]	Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi . Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506
[8]	Tahir Ullah Khan, Muhammad Adil Khan . Hermite-Hadamard inequality for new generalized conformable fractional operators. AIMS Mathematics, 2021, 6(1): 23-38. doi: 10.3934/math.2021002
[9]	Shahid Mubeen, Rana Safdar Ali, Iqra Nayab, Gauhar Rahman, Kottakkaran Sooppy Nisar, Dumitru Baleanu . Some generalized fractional integral inequalities with nonsingular function as a kernel. AIMS Mathematics, 2021, 6(4): 3352-3377. doi: 10.3934/math.2021201
[10]	Paul Bosch, Héctor J. Carmenate, José M. Rodríguez, José M. Sigarreta . Generalized inequalities involving fractional operators of the Riemann-Liouville type. AIMS Mathematics, 2022, 7(1): 1470-1485. doi: 10.3934/math.2022087

Abstract

1. Introduction

For a convex function $\sigma:\mathrm{I}\subseteq \bf{R}\to \bf{R}$ on $\mathrm{I}$ with $\nu_{1}, \nu_{2}\in \mathrm{I}$ and $\nu_{1} < \nu_{2}$ , the Hermite-Hadamard inequality is defined by ^[1]:

$\begin{equation} \sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \leq \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma(\eta)d\eta \leq \frac{\sigma(\nu_{1})+\sigma(\nu_{2})}{2}. \end{equation}$

(1.1)

The Hermite-Hadamard integral inequality (1.1) is one of the most famous and commonly used inequalities. The recently published papers ^[2,3,4] are focused on extending and generalizing the convexity and Hermite-Hadamard inequality.

The situation of the fractional calculus (integrals and derivatives) has won vast popularity and significance throughout the previous five decades or so, due generally to its demonstrated applications in numerous seemingly numerous and great fields of science and engineering ^[5,6,7].

Now, we recall the definitions of Riemann-Liouville fractional integrals.

Definition 1.1 (^[5,6,7]). Let $\sigma\in L_{1}[\nu_{1}, \nu_{2}].$ The Riemann-Liouville integrals $J_{\nu_{1}+}^{\vartheta }\sigma$ and $J_{\nu_{2}-}^{\vartheta }\sigma$ of order $\vartheta > 0$ with $\nu_{1}\geq 0$ are defined by

$\begin{equation} J_{\nu_{1}+}^{\vartheta }\sigma(x) = \frac{1}{\Gamma (\vartheta )}\int_{\nu_{1}}^{x}(x-\eta)^{\vartheta -1}\sigma(\eta)d\eta,\; \ \ \ \nu_{1} < x \end{equation}$

(1.2)

and

$\begin{equation} J_{\nu_{2}-}^{\vartheta }\sigma(x) = \frac{1}{\Gamma (\vartheta )}\int_{x}^{\nu_{2}}(\eta-x)^{\vartheta -1}\sigma(\eta)d\eta,\; \ \ x < \nu_{2}, \end{equation}$

(1.3)

respectively. Here $\Gamma (\vartheta)$ is the well-known Gamma function and $J_{\nu_{1}+}^{0}\sigma(x) = J_{\nu_{2}-}^{0}\sigma(x) = \sigma(x).$

With a huge application of fractional integration and Hermite-Hadamard inequality, many researchers in the field of fractional calculus extended their research to the Hermite-Hadamard inequality, including fractional integration rather than ordinary integration; for example see ^{[8,9,10,11,12,13,14,15,16,17,18,19,20,21]}.

In this paper, we consider the integral inequality of Hermite-Hadamard-Mercer type that relies on the Hermite-Hadamard and Jensen-Mercer inequalities. For this purpose, we recall the Jensen-Mercer inequality: Let $0 < x_{1}\leq x_{2}\leq \cdots\leq x_{n}$ and $\mu = (\mu _{1}, \mu_{2}, \ldots, \mu _{n})$ nonnegative weights such that $\sum_{i = 1}^{n}\mu_{i} = 1$ . Then, the Jensen inequality ^[22,23] is as follows, for a convex function $\sigma$ on the interval $[\nu_{1}, \nu_{2}]$ , we have

$\begin{equation} \sigma\Bigg(\sum\limits_{i = 1}^{n}\mu _{i}x_{i}\Bigg)\leq \sum\limits_{i = 1}^{n}\mu _{i}\sigma(x_{i}), \end{equation}$

(1.4)

where for all $x_{i}\in \lbrack \nu_{1}, \nu_{2}]$ and $\mu_{i}\in \lbrack 0, 1]$ , $(i = \overline{1, n})$ .

Theorem 1.1 (^[2,23]). If $\sigma$ is convex function on $[\nu_{1}, \nu_{2}]$ , then

$\begin{equation} \sigma\left(\nu_{1}+\nu_{2}-\sum\limits^{n}_{i = 1}\mu_{i}x_{i}\right)\leq \sigma(\nu_{1})+\sigma(\nu_{2})-\sum\limits^{n}_{i = 1}\mu_{i}\sigma(x_{i}), \end{equation}$

(1.5)

for each $x_{i}\in[\nu_{1}, \nu_{2}]$ and $\mu_{i}\in[0, 1]$ , $(i = \overline{1, n})$ with $\sum^{n}_{i = 1}\mu_{i} = 1$ . For some results related with Jensen-Mercer inequality, see ^[24,25,26].

In view of above indices, we establish new integral inequalities of Hermite-Hadamard-Mercer type for convex functions via the Riemann-Liouville fractional integrals in the current project. Particularly, we see that our results can cover the previous researches.

2. Main results

Theorem 2.1. For a convex function $\sigma:[\nu_{1}, \nu_{2}]\subseteq \bf{R} \to\bf{R}$ with $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

$\begin{multline} \sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\leq \frac{2^{\vartheta -1} \Gamma\left(\vartheta +1\right)}{\left( y-x\right)^{\vartheta}} \Biggl[ J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \\ +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\Biggr] \leq \sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)-\frac{\sigma\left(x\right)+\sigma\left(y\right)}{2}. \end{multline}$

(2.1)

Proof. By using the convexity of $\sigma$ , we have

$\begin{equation} \sigma\left( \nu_{1}+\nu_{2}-\frac{u+v}{2}\right) \leq \frac{1}{2}\left[ \sigma\left( \nu_{1}+\nu_{2}-u\right) +\sigma\left( \nu_{1}+\nu_{2}-v\right) \right], \end{equation}$

(2.2)

and above with $u = \frac{1-\eta}{2}x+\frac{1+\eta}{2}y$ , $v = \frac{1+\eta}{2}x+\frac{1-\eta}{2}y$ , where $x, y\in[\nu_{1}, \nu_{2}]$ and $\eta\in [0, 1]$ , leads to

$\begin{multline} \sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\leq\frac{1}{2} \Biggl[\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) \\ +\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\Biggr]. \end{multline}$

(2.3)

Multiplying both sides of (2.3) by $\eta^{\vartheta -1}$ and then integrating with respect to $\eta$ over $[0, 1]$ , we get

$\begin{align*} \frac{1}{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) &\leq \frac{1}{2}\left[\int_{0}^{1}\eta^{\vartheta-1} \sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)d\eta\right. \\ &+\left. \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)d\eta\right] \\ & = \frac{1}{2}\left[ \frac{2^{\vartheta }}{\left( y-x\right) ^{\vartheta }}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-\frac{x+y}{2}} \left( \left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)-w\right) ^{\vartheta -1}\sigma\left( w\right) dw\right. \\ &\left. +\frac{2^{\vartheta}}{\left(y-x\right)^{\vartheta}}\int_{\nu_{1}+\nu_{2}-\frac{x+y}{2}}^{\nu_{1}+\nu_{2}-x} \left( w-\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right) ^{\vartheta-1}\sigma\left( w\right) dw\right] \\ & = \frac{2^{\vartheta-1}\Gamma \left(\vartheta\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right], \end{align*}$

and thus the proof of first inequality in (2.1) is completed.

On the other hand, we have by using the Jensen-Mercer inequality:

$\begin{eqnarray} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) &\leq &\sigma\left( \nu_{1}\right) +\sigma\left( \nu_{2}\right) -\left( \frac{1-\eta}{2}\sigma\left( x\right) +\frac{1+\eta}{2}\sigma\left( y\right) \right) \end{eqnarray}$

(2.4)

$\begin{eqnarray} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) &\leq &\sigma\left( \nu_{1}\right) +\sigma\left( \nu_{2}\right) -\left( \frac{1+\eta}{2}\sigma\left( x\right) +\frac{1-\eta}{2}\sigma\left( y\right) \right). \end{eqnarray}$

(2.5)

Adding inequalities (2.4) and (2.5) to get

$\begin{multline} \sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) +\sigma\left(\nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) \\ \leq 2\left[\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)\right]-\left[\sigma\left(x\right)+\sigma\left(y\right)\right]. \end{multline}$

(2.6)

Multiplying both sides of (2.6) by $\eta^{\vartheta -1}$ and then integrating with respect to $\eta$ over $[0, 1]$ to obtain

$\begin{multline*} \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right)d\eta +\int_{0}^{1}\eta^{\vartheta-1}\sigma\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)d\eta \\ \leq \frac{2}{\vartheta}\left[\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right)\right] -\frac{1}{\vartheta }\left[ \sigma\left( x\right) +\sigma\left( y\right) \right]. \end{multline*}$

By making use of change of variables and then multiplying by $\frac{\vartheta }{2}$ , we get the second inequality in (2.1).

Remark 2.1. If we choose $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ in Theorem 2.1, then the inequality (2.1) reduces to (1.1).

Corollary 2.1. Theorem 2.1 with

● $\vartheta = 1$ becomes [24, Theorem 2.1].

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} \sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\leq\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)+\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\right] \leq \frac{\sigma\left(\nu_{1}\right)+\sigma\left(\nu_{2}\right) }{2}, \end{align*}$

which is obtained by Mohammed and Brevik in ^[10].

The following Lemma linked with the left inequality of (2.1) is useful to obtain our next results.

Lemma 2.1. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R} \to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\sigma'\in L[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( y-x\right)^{\vartheta}} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right)-}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right] \\ -\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) = \frac{\left(y-x\right)}{4}\int_{0}^{1}\eta^{\vartheta} \Bigl[\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) \\ -\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\Bigr]d\eta. \end{multline}$

(2.7)

Proof. From right hand side of (2.7), we set

$\begin{align} \varpi_{1}-\varpi_{2}&:= \int_{0}^{1}\eta^{\vartheta}\left[\sigma'\left(\nu_{1}+\nu_{2} -\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right) -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right]d\eta \\ & = \int_{0}^{1}\eta^{\vartheta }\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)d\eta \\ &-\int_{0}^{1}\eta^{\vartheta}\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right) d\eta. \end{align}$

(2.8)

By integrating by parts with $w = \nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)$ , we can deduce:

$\begin{align*} \varpi_{1}& = -\frac{2}{\left( y-x\right) }\sigma\left(\nu_{1}+\nu_{2}-y\right)+\frac{2\vartheta }{\left(y-x\right)} \int_{0}^{1}\eta^{\vartheta -1}\sigma\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) d\eta \\ & = -\frac{2}{\left(y-x\right)}\sigma\left(\nu_{1}+\nu_{2}-y\right)+\frac{2^{\vartheta+1}\vartheta}{\left(y-x\right)^{\vartheta+1}} \int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-\frac{x+y}{2}}\sigma\left(\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)-w\right)^{\vartheta -1} \sigma\left(w\right) dw \\ & = -\frac{2}{\left(y-x\right)}\sigma\left( \nu_{1}+\nu_{2}-y\right) +\frac{2^{\vartheta+1}\Gamma\left(\vartheta+1\right)}{\left( y-x\right) ^{\vartheta +1}} J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right). \end{align*}$

Similarly, we can deduce:

$\begin{equation*} \varpi_{2} = \frac{2}{y-x}\sigma\left(\nu_{1}+\nu_{2}-x\right) -\frac{2^{\vartheta+1}\Gamma\left(\vartheta+1\right)}{\left( y-x\right) ^{\vartheta +1}} J_{\left( \nu_{1}+\nu_{2}-x\right)-}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right). \end{equation*}$

By substituting $\varpi_{1}$ and $\varpi_{2}$ in (2.8) and then multiplying by $\frac{\left(y-x\right)}{4}$ , we obtain required identity (2.7).

Corollary 2.2. Lemma 2.1 with

● $\vartheta = 1$ becomes:

$\begin{multline*} \frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right) dw-\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) = \frac{\left( y-x\right) }{4}\int_{0}^{1}\eta\left[ \sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) \right. \\ \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right) \right] d\eta. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) = \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\int_{0}^{1}\eta \left[ \sigma'\left(\nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}\nu_{1}+\frac{1+\eta}{2}\nu_{2}\right) \right) \right. \\ \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}\nu_{1}+\frac{1-\eta}{2}\nu_{2}\right)\right) \right] d\eta. \end{multline*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( \nu_{2}-\nu_{1}\right)^{\vartheta }} \left[ J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) +\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right]-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \\ = \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\int_{0}^{1}\eta^{\vartheta } \left[\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}\nu_{1}+\frac{1+\eta}{2}\nu_{2}\right) \right) \right. \left. -\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}\nu_{1}+\frac{1-\eta}{2}\nu_{2}\right) \right) \right] d\eta. \end{multline*}$

Theorem 2.2. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $|\sigma'|$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \Biggl| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \Biggl[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr] \\ -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\Biggr| \leq \frac{\left( y-x\right) }{2\left( 1+\vartheta \right) }\left[ \left| \sigma'\left( \nu_{1}\right) \right| +\left| \sigma'\left(\nu_{2}\right) \right| -\frac{\left| \sigma'\left( x\right) \right| +\left| \sigma'\left( y\right) \right| }{2}\right]. \end{multline}$

(2.9)

Proof. By taking modulus of identity (2.7), we get

$\begin{multline*} \Biggl|\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] \\ -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr|\leq \frac{\left( y-x\right) }{4} \left[\int_{0}^{1}\eta^{\vartheta }\left|\sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right) \right) \right| d\eta\right. \\ \left. +\int_{0}^{1}\eta^{\vartheta }\left| \sigma'\left( \nu_{1}+\nu_{2}-\left( \frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right) \right) \right| d\eta\right]. \end{multline*}$

Then, by applying the convexity of $|\sigma'|$ and the Jensen-Mercer inequality on above inequality, we get

$\begin{align*} &\Biggl|\frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(y-x\right)^{\vartheta}} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] \\ &-\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \Biggr|\leq \frac{\left( y-x\right) }{4}\left[ \int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left( \nu_{1}\right) \right| +\left| \sigma'\left( \nu_{2}\right) \right| -\left( \frac{1+\eta}{2}\left| \sigma^{\prime }\left( x\right) \right| +\frac{1-\eta}{2}\right) \left| \sigma'\left( y\right) \right| \right] d\eta\right. \\ &\left. +\int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left(\nu_{1}\right) \right| +\left| \sigma'\left( \nu_{2}\right) \right| -\left(\frac{1-\eta}{2}\left|\sigma'\left(x\right) \right| +\frac{1+\eta}{2}\right)\left| \sigma'\left( y\right)\right| \right] d\eta\right] \\ & = \frac{\left(y-x\right)}{2\left(1+\vartheta \right)}\left[ \left|\sigma'\left( \nu_{1}\right)\right| +\left|\sigma'(\nu_{2})\right|-\frac{\left|\sigma'\left(x\right)\right|+\left|\sigma'\left(y\right)\right|}{2}\right], \end{align*}$

which completes the proof of Theorem 2.2.

Corollary 2.3. Theorem 2.2 with

● $\vartheta = 1$ becomes:

$\begin{align*} \left|\frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma(w)dw-\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right| \leq \frac{\left(y-x\right)}{4}\left[\left| \sigma'\left(\nu_{1}\right)\right|+\left|\sigma'\left(\nu_{2}\right)\right| -\frac{\left|\sigma'\left(x\right)\right|+\left|\sigma'\left(y\right)\right|}{2}\right]. \end{align*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes [27, Theorem 2.2].

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} \left\vert \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \leq \frac{\left( \nu_{2}-\nu_{1}\right) }{4}\left[ \frac{\left\vert \sigma^{\prime }\left( \nu_{1}\right) \right\vert +\left\vert \sigma^{\prime }\left( \nu_{2}\right) \right\vert }{2}\right] . \end{align*}$

Theorem 2.3. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\left| \sigma'\right|^{q}, q > 1$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left( y-x\right) ^{\vartheta }}\left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}- \frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4\sqrt[p]{\vartheta p+1}} \left[ \left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left| \sigma'\left(\nu_{2}\right)\right| ^{q} -\left(\frac{\left|\sigma'(x) \right|^{q}+3\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left(\nu_{2}\right)\right|^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right], \end{multline}$

(2.10)

where $\frac{1}{p}+\frac{1}{q} = 1$ .

Proof. By taking modulus of identity (2.7) and using Hölder's inequality, we get

$\begin{multline*} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right)^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left(\nu_{1}+\nu_{2}-x\right)-}^{\vartheta }\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta p}\right)^{\frac{1}{p}} \left\{\left(\int_{0}^{1} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)\right|^{q}d\eta\right) ^{\frac{1}{q}}\right. \\ \left. +\left(\int_{0}^{1} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right|^{q}d\eta\right) ^{\frac{1}{q}}\right\}. \end{multline*}$

Then, by applying the Jensen-Mercer inequality with the convexity of $\left|\sigma'\right|^{q}$ , we can deduce

$\begin{align*} &\left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[J_{\left(\nu_{1}+\nu_{2}-y\right)+}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ &\left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ &\leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta p}\right)^{\frac{1}{p}} \left\{ \left( \int_{0}^{1}\left| \sigma'\left( \nu_{1}\right)\right| ^{q}+\left| \sigma'\left( \nu_{2}\right) \right|^{q} -\left(\frac{1-\eta}{2}\left|\sigma'(x)\right|^{q}+\frac{1+\eta}{2}\left|\sigma'(y)\right|^{q}\right)\right)^{\frac{1}{q}}\right. \\ &\left. +\left( \int_{0}^{1}\left| \sigma'\left( \nu_{1}\right)\right| ^{q}+\left| \sigma'\left( \nu_{2}\right) \right|^{q} -\left(\frac{1+\eta}{2}\left| \sigma'\left( x\right)\right|^{q}+\frac{1-\eta}{2}\left| \sigma'\left( y\right) \right|^{q} \right) \right)^{\frac{1}{q}}\right\} \\ & = \frac{\left(y-x\right)}{4\sqrt[p]{\vartheta p+1}}\left[\left( \left| \sigma'\left( \nu_{1}\right) \right| ^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left|\sigma'\left( x\right) \right|^{q} +3\left| \sigma'\left(y\right) \right|^{q} }{4}\right) \right) ^{\frac{1}{q}}\right. \\ &\left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left| \sigma'\left( \nu_{2}\right) \right| ^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right], \end{align*}$

which completes the proof of Theorem 2.3.

Corollary 2.4. Theorem 2.3 with

● $\vartheta = 1$ becomes:

$\begin{multline*} \left|\frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right)dw -\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right| \\ \leq \frac{\left( y-x\right)}{4\sqrt[p]{p+1}}\left[\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left|\sigma'\left( x\right) \right|^{q} +3\left|\sigma'\left(y\right) \right|^{q}}{4}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left(\left|\sigma'\left( \nu_{1}\right)\right|^{q}+\left| \sigma'\left( \nu_{2}\right)\right|^{q} -\left(\frac{3\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q} }{4}\right)\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} &&\left\vert \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw-\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \leq \frac{\left( \nu_{2}-\nu_{1}\right) }{2^{\frac{2}{p}}}\left(\frac{1}{p+1}\right)^{\frac{1}{p}}\left[ \left\vert \sigma^{\prime }\left( \nu_{1}\right) \right\vert +\left\vert \sigma^{\prime }\left( \nu_{2}\right) \right\vert \right] . \end{align*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{align*} &\left\vert \frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[ J_{\nu_{1}+}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right. \right. \left. \left. +\; J_{\nu_{2}-}^{\vartheta }\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right] -\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right) \right\vert \\ &\leq \frac{2^{\vartheta-1-\frac{2}{q}}}{\nu_{2}-\nu_{1}}\left(\frac{1}{p+1}\right)^{\frac{1}{p}}\left[ \left\vert \sigma'\left( \nu_{1}\right) \right\vert +\left\vert \sigma'\left( \nu_{2}\right) \right\vert \right]. \end{align*}$

Theorem 2.4. Let $\sigma:[\nu_{1}, \nu_{2}]\subseteq\bf{R}\to\bf{R}$ be a differentiable function on $(\nu_{1}, \nu_{2})$ and $\left|\sigma'\right|^{q}, q\geq 1$ is convex on $[\nu_{1}, \nu_{2}]$ with $\nu_{1}\leq \nu_{2}$ and $x, y\in[\nu_{1}, \nu_{2}]$ . Then, we have:

$\begin{multline} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right.\right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left(y-x\right)}{4\left(\vartheta+1\right)}\left[\left(\left|\sigma'(\nu_{1})\right|^{q} +\left|\sigma'\left(\nu_{2}\right)\right|^{q}-\left( \frac{\left| \sigma'\left(x\right)\right|^{q} +\left( 2\vartheta+3\right)\left|\sigma'(y)\right|^{q}}{2\left(\vartheta+2\right) }\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left|\sigma'\left(\nu_{1}\right)\right|^{q}+\left| \sigma'\left( \nu_{2}\right)\right|^{q} -\left(\frac{\left(2\vartheta+3\right)\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{2\left( \vartheta+2\right) }\right) \right) ^{\frac{1}{q}}\right]. \end{multline}$

(2.11)

Proof. By taking modulus of identity (2.7) with the well-known power mean inequality, we can deduce

$\begin{multline*} \left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right) ^{\vartheta }} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right) +}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ \left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{4}\left( \int_{0}^{1}\eta^{\vartheta }\right) ^{1-\frac{1}{q}} \left\{\left(\int_{0}^{1}\eta^{\vartheta} \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1-\eta}{2}x+\frac{1+\eta}{2}y\right)\right)\right|^{q}d\eta\right)^{\frac{1}{q}}\right. \\ \left. +\left( \int_{0}^{1}\eta^{\vartheta } \left|\sigma'\left(\nu_{1}+\nu_{2}-\left(\frac{1+\eta}{2}x+\frac{1-\eta}{2}y\right)\right)\right|^{q}d\eta\right)^{\frac{1}{q}}\right\} . \end{multline*}$

By applying the Jensen-Mercer inequality with the convexity of $\left|\sigma'\right|^{q}$ , we can deduce

$\begin{align*} &\left| \frac{2^{\vartheta -1}\Gamma \left( \vartheta +1\right) }{\left(y-x\right)^{\vartheta}} \left[ J_{\left( \nu_{1}+\nu_{2}-y\right)+}^{\vartheta}\sigma\left(\nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right. \right. \\ &\left. \left. +\; J_{\left( \nu_{1}+\nu_{2}-x\right) -}^{\vartheta }\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right)\right] -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ &\leq \frac{\left( y-x\right)}{4}\left(\int_{0}^{1}\eta^{\vartheta }\right)^{1-\frac{1}{q}} \left\{ \left( \int_{0}^{1}\eta^{\vartheta }\left[ \left|\sigma'\left( \nu_{1}\right) \right|^{q} +\left| \sigma'\left(\nu_{2}\right) \right| ^{q}-\left( \frac{1-\eta}{2}\left| \sigma'\left(x\right)\right|^{q} +\frac{1+\eta}{2}\left| \sigma'\left( y\right)\right| ^{q}\right) \right] \right) ^{\frac{1}{q}}\right. \\ &\left. +\left( \int_{0}^{1}\eta^{\vartheta }\left[ \left| \sigma'\left(\nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right) \right|^{q}-\left( \frac{1+\eta}{2}\left| \sigma'\left( x\right) \right|^{q} +\frac{1-\eta}{2}\left| \sigma'\left( y\right) \right|^{q}\right) \right] \right) ^{\frac{1}{q}}\right\} \\ & = \frac{\left( y-x\right) }{4\left( \vartheta +1\right) }\left[\left(\left|\sigma'\left(\nu_{1}\right)\right|^{q} +\left|\sigma'\left( \nu_{2}\right) \right| ^{q}-\left( \frac{\left| \sigma'\left(x\right) \right|^{q} +\left(2\vartheta +3\right)\left|\sigma'(y)\right|^{q}}{2\left(\vartheta+2\right)}\right)\right)^{\frac{1}{q}}\right. \\ &\left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left( \nu_{2}\right)\right|^{q} -\left( \frac{\left( 2\vartheta +3\right) \left| \sigma'\left( x\right) \right|^{q} +\left|\sigma'\left( y\right)\right|^{q}}{2\left(\vartheta+2\right)}\right)\right)^{\frac{1}{q}}\right], \end{align*}$

which completes the proof of Theorem 2.4.

Corollary 5. Theorem 2.4 with

● $q = 1$ becomes Theorem 2.2.

● $\vartheta = 1$ becomes:

$\begin{multline*} \left| \frac{1}{y-x}\int_{\nu_{1}+\nu_{2}-y}^{\nu_{1}+\nu_{2}-x}\sigma\left(w\right) dw -\sigma\left( \nu_{1}+\nu_{2}-\frac{x+y}{2}\right) \right| \\ \leq \frac{\left( y-x\right) }{8}\left[ \left( \left|\sigma'\left( \nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right)\right|^{q}-\left( \frac{\left| \sigma'\left( x\right)\right|^{q} +5\left|\sigma'\left(y\right)\right|^{q}}{6}\right)\right)^{\frac{1}{q}}\right. \\ \left. +\left( \left| \sigma'\left( \nu_{1}\right) \right|^{q}+\left|\sigma'\left(\nu_{2}\right)\right|^{q} -\left(\frac{5\left|\sigma'(x)\right|^{q}+\left|\sigma'(y)\right|^{q}}{6}\right)\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $\vartheta = 1$ , $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \left| \frac{1}{\nu_{2}-\nu_{1}}\int_{\nu_{1}}^{\nu_{2}}\sigma\left( w\right) dw -\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)\right| \\ \leq \frac{\left(y-x\right)}{8}\left[\left(\frac{5\left|\sigma'\left(\nu_{1}\right)\right|^{q} +\left| \sigma'\left( \nu_{2}\right)\right| ^{q}}{6}\right)^{\frac{1}{q}} +\left(\frac{\left|\sigma'(\nu_{1})\right|^{q}+5\left| \sigma'(\nu_{2})\right|^{q}}{6}\right)^{\frac{1}{q}}\right]. \end{multline*}$

● $x = \nu_{1}$ and $y = \nu_{2}$ becomes:

$\begin{multline*} \left| \frac{2^{\vartheta-1}\Gamma\left(\vartheta+1\right)}{\left(\nu_{2}-\nu_{1}\right)^{\vartheta}} \left[J_{\nu_{1} +}^{\vartheta}\sigma\left(\frac{\nu_{1}+\nu_{2}}{2}\right)+\; J_{\nu_{2} -}^{\vartheta }\sigma\left( \frac{\nu_{1}+\nu_{2}}{2}\right)\right] -\sigma\left( \frac{\nu_{1}+\nu_{2}}{2}\right)\right| \\ \leq \frac{\left(\nu_{2}-\nu_{1}\right)}{4\left(\vartheta+1\right)} \left[\left(\frac{\left(2\vartheta+3\right)\left|\sigma'(\nu_{1})\right|^{q}+\left|\sigma'(\nu_{2})\right|^{q}}{2\left(\vartheta+2\right)}\right)^{\frac{1}{q}} +\left(\frac{\left|\sigma'(\nu_{1})\right|^{q}+\left(2\vartheta+3\right)\left|\sigma'(\nu_{2})\right|^{q}}{2\left(\vartheta+2\right)}\right) ^{\frac{1}{q}}\right]. \end{multline*}$

3. Applications to special means

Here, we consider the following special means:

● The arithmetic mean:

$\mathcal{A}(\nu_{1},\nu_{2}) = \frac{\nu_{1}+\nu_{2}}{2}, \quad \nu_{1}, \nu_{2}\geq 0.$

● The harmonic mean:

$\mathcal{H}(\nu_{1},\nu_{2}) = \frac{2\nu_{1}\nu_{2}}{\nu_{1}+\nu_{2}}, \quad \nu_{1}, \nu_{2} > 0.$

● The logarithmic mean:

$\begin{equation*} \mathcal{L}(\nu_{1},\nu_{2}) = \begin{cases} \frac{\nu_{2}-\nu_{1}}{\ln \nu_{2}-\ln \nu_{1}}, & \text{if}\; \nu_{1}\neq \nu_{2}, \\ \nu_{1}, & \text{if}\; \nu_{1} = \nu_{2}, \end{cases} \quad \nu_{1}, \nu_{2} > 0. \end{equation*}$

● The generalized logarithmic mean:

$\begin{equation*} \mathcal{L}_{n}(\nu_{1},\nu_{2}) = \begin{cases} \left[\frac{\nu_{2}^{n+1}-\nu_{1}^{n+1}}{\left(n+1\right)(\nu_{2}-\nu_{1})}\right]^{\frac{1}{n}}, & \text{if}\; \nu_{1}\neq \nu_{2} \\ \nu_{1}, & \text{if}\; \nu_{1} = \nu_{2}, \end{cases} \quad \nu_{1},\nu_{2} > 0;\; n\in\bf{Z}\setminus \{-1,0\}. \end{equation*}$

Proposition 3.1. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

(3.1)

Proof. By applying Corollary 2.3 (first item) for the convex function $\sigma(x) = x^{n}, \, x > 0$ , one can obtain the result directly.

Proposition 3.2. Let $0 < \nu_{1} < \nu_{2}$ . Then, for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

(3.2)

Proof. By applying Corollary 2.3 (first item) for the convex function $\sigma(x) = \frac{1}{x}, \, x > 0$ , one can obtain the result directly.

Proposition 3.3. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, we have:

$\begin{equation} \left| \mathcal{L}_{n}^{n}(\nu_{1},\nu_{2}) -\mathcal{A}^{n}(\nu_{1},\nu_{2}) \right| \leq \frac{n(\nu_{2}-\nu_{1})}{4} \left[ \mathcal{A}\left(\nu_{1}^{n-1}, \nu_{2}^{n-1}\right) \right], \end{equation}$

(3.3)

and

$\begin{equation} \left| \mathcal{L}^{-1}(\nu_{1},\nu_{2}) -\mathcal{A}^{-1}(\nu_{1},\nu_{2}) \right| \leq \frac{(\nu_{2}-\nu_{1})}{4}\mathcal{H}^{-1}\left( \nu_{1}^{2},\nu_{2}^{2}\right) . \end{equation}$

(3.4)

Proof. By setting $x = \nu_{1}$ and $y = \nu_{2}$ in results of Proposition 3.1 and Proposition 3.2, one can obtain the Proposition 3.3.

Proposition 3.4. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for $q > 1, \, \frac{1}{p}+\frac{1}{q} = 1$ and for all $x, y\in[\nu_{1}, \nu_{2}],$ we have:

$\begin{multline} \left| \mathcal{L}_{n}^{n}\left( \nu_{1}+\nu_{2}-y,\nu_{1}+\nu_{2}-x\right) -\left( 2\mathcal{A}(\nu_{1},\nu_{2}) -\mathcal{A}\left( x,y\right) \right) ^{n}\right| \\ \leq \frac{n(y-x)}{4\sqrt[p]{p+1}}\Bigg\{\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(x^{q(n-1)}, 3y^{q(n-1)}\right) \right]^{\frac{1}{q}} \\ +\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(3x^{q(n-1)}, y^{q(n-1)}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.5)

Proof. By applying Corollary 2.4 (first item) for convex function $\sigma\left(x\right) = x^{n}, \, x > 0$ , one can obtain the result directly.

Proposition 3.5. Let $0 < \nu_{1} < \nu_{2}$ . Then, for $q > 1, \, \frac{1}{p}+\frac{1}{q} = 1$ and for all $x, y\in[\nu_{1}, \nu_{2}]$ , we have:

$\begin{multline} \left| \mathcal{L}^{-1}\left( \nu_{1}+\nu_{2}-y,\nu_{1}+\nu_{2}-x\right) -\left( 2\mathcal{A}(\nu_{1},\nu_{2}) -\mathcal{A}\left(x,y\right)\right)^{-1}\right| \\ \leq \frac{\sqrt[q]{2}(y-x)}{4\sqrt[p]{p+1}}\Bigg\{\left[ \mathcal{H}^{-1}\left(\nu_{1}^{2q},\nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(x^{2q}, 3y^{2q}\right) \right]^{\frac{1}{q}} \\ +\left[\mathcal{H}^{-1}\left(\nu_{1}^{2q},\nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(3x^{2q}, y^{2q}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.6)

Proof. By applying Corollary 2.4 (first item) for the convex function $\sigma\left(x\right) = \frac{1}{x}, \, x > 0$ , one can obtain the result directly.

Proposition 3.6. Let $0 < \nu_{1} < \nu_{2}$ and $n\in\bf{N}$ , $n \geq 2$ . Then, for $q > 1$ and $\frac{1}{p}+\frac{1}{q} = 1$ , we have:

$\begin{multline} \left| \mathcal{L}_{n}^{n}(\nu_{1},\nu_{2}) -\mathcal{A}^{n}(\nu_{1},\nu_{2}) \right| \leq \frac{n(\nu_{2}-\nu_{1})}{4\sqrt[p]{p+1}} \Bigg\{\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left( \nu_{1}^{q(n-1)}, 3\nu_{2}^{q(n-1)}\right) \right]^{\frac{1}{q}} \\ +\left[ 2\mathcal{A}\left( \nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) -\frac{1}{2}\mathcal{A}\left(3\nu_{1}^{q(n-1)}, \nu_{2}^{q(n-1)}\right) \right]^{\frac{1}{q}}\Bigg\}, \end{multline}$

(3.7)

and

$\begin{multline} \left| \mathcal{L}^{-1}(\nu_{1},\nu_{2}) -\mathcal{A}^{-1}(\nu_{1},\nu_{2}) \right| \leq \frac{\sqrt[q]{2}(\nu_{2}-\nu_{1})}{4\sqrt[p]{p+1}} \Bigg\{\left[ \mathcal{H}^{-1}\left( \nu_{1}^{2q}, \nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(\nu_{1}^{2q}, 3\nu_{2}^{2q}\right) \right]^{\frac{1}{q}} \\ +\left[ \mathcal{H}^{-1}\left( \nu_{1}^{2q}, \nu_{2}^{2q}\right) -\frac{3}{4}\mathcal{H}^{-1}\left(3\nu_{1}^{2q}, \nu_{2}^{2q}\right) \right]^{\frac{1}{q}}\Bigg\}. \end{multline}$

(3.8)

Proof. By setting $x = \nu_{1}$ and $y = \nu_{2}$ in results of Proposition 3.4 and Proposition 3.5, one can obtain the Proposition 3.6.

4. Conclusions

As we emphasized in the introduction, integral inequality is the most important field of mathematical analysis and fractional calculus. By using the well-known Jensen-Mercer and power mean inequalities, we have proved new inequalities of Hermite-Hadamard-Mercer type involving Riemann-Liouville fractional operators. In the last section, we have considered some propositions in the context of special functions; these confirm the efficiency of our results.

Acknowledgements

We would like to express our special thanks to the editor and referees. Also, the first author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	U. Ahmad, S. Ahmed, Muhammad Javaid, M. N. Alam, Computing Fault-tolerant Metric Dimension of Connected Graphs, J. Math., 2022 (2022), Article ID 9773089. https://doi.org/10.1155/2022/97730892022 doi: 10.1155/2022/97730892022
[2]	R. C. Brigham, G. Chartrand, R. D. Dutton, P. Zhang, Resolving domination in graphs, Math. Bohem., 128 (2003), 325–364. https://doi.org/10.21136/MB.2003.134179 doi: 10.21136/MB.2003.134179
[3]	G. Chartrand, L. Eroh, M. A. Johnson, O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 (2000), 99–113.
[4]	G. Chartrand, V. Saenpholphat, P. Zhang, The independent resolving number of a graph, Math. Bohem., 128 (2003), 379–393. https://doi.org/10.21136/MB.2003.134003 doi: 10.21136/MB.2003.134003
[5]	A. Estrada-Moreno, J. A. Rodriguez-Velazquez, I. G. Yero, The k-metric dimension of a graph, Appl. Math. Inf. Sci., 9 (2015), 2829–2840.
[6]	M. M. Danasa, J. Kraticab, A. Savic, Z. Maksimovic, Some new general lower bounds for mixed metric dimension of graphs, Filomat, 35 (2021). https://doi.org/10.2298/FIL2113275M doi: 10.2298/FIL2113275M
[7]	F. Harary, R. A. Melter, On the metric dimension of a graph, Ars Combinatoria, 2 (1976), 191–195.
[8]	A. Kelenc, N. Tratnik, I. G. Yero, Uniquely identifying the edges of a graph: the edge metric dimension, Discrete Appl. Math., 251 (2018), 204–220. https://doi.org/10.1016/j.dam.2018.05.052 doi: 10.1016/j.dam.2018.05.052
[9]	A. Kelenc, D. Kuzia, A. Taranenko, I. G. Yero, Mixed metric dimension of graphs, Appl. Math. Comput., 314 (2017), 429–438. https://doi.org/10.1016/j.amc.2017.07.027 doi: 10.1016/j.amc.2017.07.027
[10]	S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70 (1996), 217–229.
[11]	O. R. Oellermann, J. Peters-Fransen, The strong metric dimension of graphs and digraphs, Discret Appl. Math., 155 (2007), 356–364. https://doi.org/10.1016/j.dam.2006.06.009 doi: 10.1016/j.dam.2006.06.009
[12]	F. Okamoto, B. Phinezyn, P. Zhang, The local metric dimension of a graph, Math. Bohem., 135 (2010), 239–255. https://doi.org/10.21136/MB.2010.140702 doi: 10.21136/MB.2010.140702
[13]	H. Razza, Y. Ji, Computing the Mixed Metric Dimension of a Generalized Petersen Graph $P(n, 2)$ , Front. Phys., 28 July 2020. https://doi.org/10.3389/fphy.2020.00211
[14]	P. J. Slater, Leaves of trees, Proceeding of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, 14 (1975), 549–559.
[15]	J. Sedlar, R. Škrekovsk, Mixed metric dimension of graphs with edge disjoint cycles, Discrete Appl. Math., 300 (2021), 1–8. https://doi.org/10.1016/j.dam.2021.05.004 doi: 10.1016/j.dam.2021.05.004
[16]	G. Sudhakara, A. R. Hemanth Kumar, Graphs with metric dimension two a characterization, Adv. Appl. Discrete Math., 4 (2009), 169–186.

This article has been cited by:

1.	Tariq A. Aljaaidi, Deepak B. Pachpatte, Ram N. Mohapatra, The Hermite–Hadamard–Mercer Type Inequalities via Generalized Proportional Fractional Integral Concerning Another Function, 2022, 2022, 1687-0425, 1, 10.1155/2022/6716830
2.	Saad Ihsan Butt, Ahmet Ocak Akdemir, Muhammad Nadeem, Nabil Mlaiki, İşcan İmdat, Thabet Abdeljawad, $(m, n)$ -Harmonically polynomial convex functions and some Hadamard type inequalities on the co-ordinates, 2021, 6, 2473-6988, 4677, 10.3934/math.2021275
3.	Ifra Bashir Sial, Nichaphat Patanarapeelert, Muhammad Aamir Ali, Hüseyin Budak, Thanin Sitthiwirattham, On Some New Ostrowski–Mercer-Type Inequalities for Differentiable Functions, 2022, 11, 2075-1680, 132, 10.3390/axioms11030132
4.	Deniz Uçar, Inequalities for different type of functions via Caputo fractional derivative, 2022, 7, 2473-6988, 12815, 10.3934/math.2022709
5.	Soubhagya Kumar Sahoo, Y.S. Hamed, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Kamsing Nonlaopon, New midpoint type Hermite-Hadamard-Mercer inequalities pertaining to Caputo-Fabrizio fractional operators, 2023, 65, 11100168, 689, 10.1016/j.aej.2022.10.019
6.	Muhammad Imran Asjad, Waqas Ali Faridi, Mohammed M. Al-Shomrani, Abdullahi Yusuf, The generalization of Hermite-Hadamard type Inequality with exp-convexity involving non-singular fractional operator, 2022, 7, 2473-6988, 7040, 10.3934/math.2022392
7.	Churong Chen, Discrete Caputo Delta Fractional Economic Cobweb Models, 2023, 22, 1575-5460, 10.1007/s12346-022-00708-5
8.	Soubhagya Kumar Sahoo, Ravi P. Agarwal, Pshtiwan Othman Mohammed, Bibhakar Kodamasingh, Kamsing Nonlaopon, Khadijah M. Abualnaja, Hadamard–Mercer, Dragomir–Agarwal–Mercer, and Pachpatte–Mercer Type Fractional Inclusions for Convex Functions with an Exponential Kernel and Their Applications, 2022, 14, 2073-8994, 836, 10.3390/sym14040836
9.	Muhammad Tariq, Sotiris K. Ntouyas, Asif Ali Shaikh, A Comprehensive Review of the Hermite–Hadamard Inequality Pertaining to Fractional Integral Operators, 2023, 11, 2227-7390, 1953, 10.3390/math11081953
10.	Loredana Ciurdariu, Eugenia Grecu, Hermite–Hadamard–Mercer-Type Inequalities for Three-Times Differentiable Functions, 2024, 13, 2075-1680, 413, 10.3390/axioms13060413
11.	Muhammad Aamir Ali, Thanin Sitthiwirattham, Elisabeth Köbis, Asma Hanif, Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications, 2024, 13, 2075-1680, 114, 10.3390/axioms13020114
12.	Muhammad Aamir Ali, Christopher S. Goodrich, On some new inequalities of Hermite–Hadamard–Mercer midpoint and trapezoidal type in q-calculus, 2024, 44, 0174-4747, 35, 10.1515/anly-2023-0019
13.	Thanin Sitthiwirattham, Ifra Sial, Muhammad Ali, Hüseyin Budak, Jiraporn Reunsumrit, A new variant of Jensen inclusion and Hermite-Hadamard type inclusions for interval-valued functions, 2023, 37, 0354-5180, 5553, 10.2298/FIL2317553S
14.	Muhammad Aamir Ali, Zhiyue Zhang, Michal Fečkan, GENERALIZATION OF HERMITE–HADAMARD–MERCER AND TRAPEZOID FORMULA TYPE INEQUALITIES INVOLVING THE BETA FUNCTION, 2024, 54, 0035-7596, 10.1216/rmj.2024.54.331
15.	Bahtiyar Bayraktar, Péter Kórus, Juan Eduardo Nápoles Valdés, Some New Jensen–Mercer Type Integral Inequalities via Fractional Operators, 2023, 12, 2075-1680, 517, 10.3390/axioms12060517
16.	THANIN SITTHIWIRATTHAM, MIGUEL VIVAS-CORTEZ, MUHAMMAD AAMIR ALI, HÜSEYIN BUDAK, İBRAHIM AVCI, A STUDY OF FRACTIONAL HERMITE–HADAMARD–MERCER INEQUALITIES FOR DIFFERENTIABLE FUNCTIONS, 2024, 32, 0218-348X, 10.1142/S0218348X24400164
17.	Muhammad Ali, Hüseyin Budak, Elisabeth Köbis, Some new and general versions of q-Hermite-Hadamard-Mercer inequalities, 2023, 37, 0354-5180, 4531, 10.2298/FIL2314531A

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1538) PDF downloads(80) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(3)

AIMS Mathematics

Graphs with mixed metric dimension three and related algorithms

Related Papers:

Abstract

1. Introduction

2. Main results

3. Applications to special means

4. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Mathematics

Graphs with mixed metric dimension three and related algorithms

Related Papers:

Abstract

1. Introduction

2. Main results

3. Applications to special means

4. Conclusions

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog