The functional roles of T-cadherin in mammalian biology

Jade Sternberg; Miriam Wankell; V. Nathan Subramaniam; Lionel W. Hebbard; Jade Sternberg; Miriam Wankell; V. Nathan Subramaniam; Lionel W. Hebbard

doi:10.3934/molsci.2017.1.62

AIMS Molecular Science

2017, Volume 4, Issue 1: 62-81. doi: 10.3934/molsci.2017.1.62

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Review Topical Sections

The functional roles of T-cadherin in mammalian biology

1.
College of Public Health, Medical and Veterinary Sciences, James Cook University, Townsville, Australia
2.
Comparative Genomics Centre, James Cook University, Townsville, Australia
3.
Australian Institute of Tropical Health and Medicine, James Cook University, Townsville, Australia
4.
Department of Molecular and Cell Biology, James Cook University, Townsville, Australia
5.
Institute of Health and Biomedical Innovation and School of Biomedical Sciences, Queensland University of Technology (QUT), Brisbane. Australia

Received: 21 November 2016 Accepted: 26 January 2017 Published: 13 February 2017

T-cadherin is a cadherin and cell adhesion molecule that is anchored to the cell surface membrane through a glycosylphosphatidylinositol moiety. T-cadherin lacks a transmembrane and cytoskeletal domain, suggesting that it must interact with other membrane-bound molecules to elicit cellular signaling to modulate normal cellular functions, and alternatively its absence can be a factor in promoting neoplasia. Moreover, apart from binding to itself it can sequester adiponectin to the cell surface. Consistent with these observations, recent research has expanded the scope of T-cadherin’s role in cancer, neuronal function, metabolism and cardiovascular disease. In this context, we highlight the experimental and genomic evidence that links T-cadherin with these diseases. In particular, we discuss how T-cadherin homophilic and heterophilic interactions impact on signaling pathways and cellular behavior.

Keywords:

Citation: Jade Sternberg, Miriam Wankell, V. Nathan Subramaniam, Lionel W. Hebbard. The functional roles of T-cadherin in mammalian biology[J]. AIMS Molecular Science, 2017, 4(1): 62-81. doi: 10.3934/molsci.2017.1.62

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Abstract

1. Introduction

In recent years, convexity theory has gained special attention by many researchers because of it engrossing properties and expedient characterizations. It has many applications in fields like biology, numerical analysis and statistics (see ^[1,2,3,4]). Mathematical inequalities are extensively studied with all type of convex functions (see^{[1,3,11,13,14,16]}). One of the fundamental inequality is Hermite-Hadamard inequality. It has been discussed via different types of convexities and became the center of attention for many researchers. Recently, in 2016, Khan et al. have discussed generalizations of Hermite-Hadamard type for $MT$ -convex functions ^[26]. In 2017, Khan et al. studied some new inequalities of Hermite-Hadamard types ^[27]. In 2019, Khurshid et al. have utilized conformable fractional integrals via preinvex functions ^[28]. In 2020, Khan et al. have discussed Hermite-Hadamard type inequalities via quantum calculus involving green function ^[29], Mohammed et al. have established a new version of Hermite-Hadamard inequality for Riemann-Liouville fractional integrals ^[30], Han et al. used fractional integral to generalize Hermite-Hadamard inequality for convex functions ^[31], Zhao et al. utilized harmonically convex functions to generalized fractional integral inequalities of Hermite-Hdamrd type ^[32], Awan et al. presented new inequalities of Hermite-Hdamard type for $n$ -polynomial harmonically convex functions ^[33]. In 2022, Khan et al. introduced some new versions of Hermite-Hadamard integral inequalities in fuzzy fractional calculus for generalized pre-invex functions via fuzzy-interval-valued settings ^[34]. This reflects the importance of Hermite Hadamard type inequalities among current research.

In ^[9], $s$ -convex function is given as,

Definition 1.1. A real valued function $\chi$ is called $s$ -convex function on $\mathbb{R}$ , if

$\begin{equation*} \chi \Big(\varsigma \rho+ (1-\varsigma)\gamma\Big)\leq \varsigma^{s} \chi(\rho)+(1-\varsigma)^{s} \chi(\gamma), \end{equation*}$

for each $\rho, \gamma \in \mathbb{R}$ and $\varsigma \in (0, 1)$ where $s\in (0, 1]$ .

In ^[10], $m$ -convexity is discussed as,

Definition 1.2. A real valued function $\chi$ defined on $[0, b]$ is said to be a $m$ -convex function for $m \in [0, 1]$ , if

$\begin{equation*} \chi \Big(\varsigma \rho+ m(1-\varsigma)\gamma\Big)\leq \varsigma\chi(\rho)+m(1-\varsigma) \chi(\gamma), \end{equation*}$

holds for all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$ .

$(s, m)$ -convexity in ^[17] is discussed as,

Definition 1.3. A function $\chi: [0, b] \longrightarrow \mathbb{R}$ , $b > 0$ is said to be a $(s, m)$ -convex function in the second sense where $s, m \in (0, 1]^{2}$ , if

$\begin{equation*} \chi \Big(\varsigma \rho+ m(1-\varsigma)\gamma\Big)\leq \varsigma^{s}\chi(\rho)+m(1-\varsigma)^{s} \chi(\gamma), \end{equation*}$

holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$ .

Equivalent definition for $(s, m)$ –convex functions:

Let $\rho, \alpha, \gamma \in [0, b]$ , $\rho < \alpha < \gamma$

$\begin{equation} \chi (\alpha) \le {\left(\frac{{\gamma - \alpha}}{{\gamma - \rho}}\right)^s}\chi (\rho) + {m\left(\frac{{\alpha - \rho}}{{\gamma - \rho}}\right)^{s}}\chi (\gamma). \end{equation}$

(1.1)

Hölder-İşcan Inequality ^[5]:

Let $p > 1$ , $\chi$ and $\psi$ be real valued functions defined on $\left[\rho, \gamma \right]$ and $|\chi|^{p}, |\psi|^{q}$ are integrable functions on interval $\left[\rho, \gamma \right]$

$\begin{equation} \begin{array}{l} \int_\rho ^\gamma | \chi (\omega)\psi (\omega)|d\omega \le \frac{1}{{\gamma - \rho }}{\left( {\int_\rho ^\gamma {(\gamma - \omega)} |\chi (\omega){|^p}d\omega} \right)^{\frac{1}{p}}}{\left( {\int_\rho ^\gamma {(\gamma - \omega)} |\psi (\omega){|^q}d\omega} \right)^{\frac{1}{q}}} \\ + \frac{1}{{\gamma - \rho }}{\left( {\int_\rho^\gamma {(\omega - \rho )} |\chi (\omega){|^p}d\omega} \right)^{\frac{1}{p}}}{\left( {\int_\rho ^\gamma {(\omega - \rho )} |\psi (\omega){|^q}d\omega} \right)^{\frac{1}{q}}}, \end{array} \end{equation}$

(1.2)

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

Following lemma is useful to obtain our main results.

Lemma 1.4. ^[8] For $n\in \mathbb{N}$ , let $\chi:U\subseteq \mathbb{R} \longrightarrow \mathbb{R}$ be $n$ -times differentiable mapping on $U^{\circ}$ , where $\rho, \gamma \in U^{\circ}$ , $\rho < \gamma$ and $\chi^{n} \in L[\rho, \gamma]$ , we have following identity

$\begin{equation} \sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}}\left (\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}\right) - \int\limits_\rho ^\gamma {\chi (\omega)d\omega} = \frac{{{{( - 1)}^{n + 1}}}}{{n!}}\int\limits_\rho ^\gamma {{\omega^n}} {\chi ^{(n)}}(\omega)d\omega, \end{equation}$

(1.3)

where an empty set is understood to be nil.

In this paper, Hölder-İşcan inequality is used to modify inequalities involving functions having $s$ -convex or $s$ -concave derivatives at certain powers. The purpose of this paper is to establish some generalized inequalities for $n$ -times differentiable $(s, m)$ -convex functions. Applications of these inequalities to means are also discussed. Means are defined as,

Let $0 < \rho < \gamma$ ,

$\begin{equation*} A(\rho, \gamma) = \frac{\rho+\gamma}{2}, \end{equation*}$

$\begin{equation*} G(\rho, \gamma) = \sqrt{\rho\gamma}, \end{equation*}$

$\begin{equation*} L_{p}(\rho, \gamma) = \Big (\frac{\gamma^{p+1}-\rho^{p+1}}{(p+1)(\gamma-\rho)}\Big )^{\frac{1}{p}}, \end{equation*}$

where $p\neq0, -1$ and $\rho\neq\gamma$ .

2. Main results

2.1. Modified inequalities for $n$ -times differentiable convex functions

Theorem 2.1. For any positive integer $n$ , let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be $n$ -times differentiable mapping on $U^{\circ}$ , where $\rho, \gamma \in U^{\circ}$ with $\rho < \gamma$ . If $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$ is $(s, m)$ -convex on interval $[\rho, \gamma]$ then

$\begin{equation} \begin{array}{l} \left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}} \left(\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}\right) - \int\limits_\rho ^\mu {\chi (\omega)d\omega} } \right|\\ \le \frac{1}{{n!}}{(\gamma-\rho)^{\frac{1}{q}}}\left( \begin{array}{l} {[\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )]^{\frac{1}{p}}}{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)(s + 1)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 2)}}} \right]^{\frac{1}{q}}}\\ + {[L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )]^{\frac{1}{p}}}{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 1)(s + 2)}}} \right]^{\frac{1}{q}}} \end{array} \right), \end{array} \end{equation}$

(2.1)

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

Proof. Since $|\chi^{n}|^q$ is $(s, m)$ -convex by using inequality (1.1) for $\rho < \omega < \gamma$ , using Lemma 1.4 and Hölder-Işcan inequality (1.2),

$\begin{equation*} \begin{array}{c} {\left| {{\chi ^n}(\omega)} \right|^q} \le {\left| {{\chi ^n}(\frac{{\omega - \rho }}{{\gamma - \rho }}\gamma +m \frac{{\gamma - \omega}}{{\gamma - \rho }}\rho )} \right|^q}\\ \\ \le {(\frac{{\omega - \rho }}{{\gamma - \rho }})^s}{\left| {{\chi ^n}(\gamma )} \right|^q} + {m(\frac{{\gamma - \omega}}{{\gamma - \rho }})^{s}}{\left| {{\chi ^n}(\rho )} \right|^q}, \\ \begin{array}{l} \left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu }\left( {\frac{{{\chi ^{(\nu )}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu )}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}} \right) - \int\limits_\rho ^\gamma {\chi (\omega )d\omega } } } \right|\\ \le \frac{1}{{n!}}\int\limits_\rho ^\gamma {{\omega ^n}} \left| {{\chi ^{(n)}}(\omega )} \right|d\omega , \end{array}\\ \le \frac{1}{{n!}}\frac{1}{{\gamma - \rho }}\left\{ \begin{array}{l} {\left( {\int\limits_\rho ^\gamma {(\gamma - \omega ){\omega ^{np}}d\omega } } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega ){{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega} } \right)^{\frac{1}{q}}} + \\ {\left( {\int\limits_\rho ^\gamma {(\omega - \rho ){\omega ^{np}}d\omega } } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^\gamma {(\omega - \rho ){{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega } } \right)^{\frac{1}{q}}} \end{array} \right\},\\ \le \frac{1}{{n!}}\frac{1}{{\gamma - \rho }}{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega){\omega^{np}}d\omega} } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega)\left[ {{{(\frac{{\omega - \rho }}{{\gamma - \rho }})}^s}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + {{m(\frac{{\gamma - \omega}}{{\gamma - \rho }})}^s}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]d\omega} } \right)^{\frac{1}{q}}}\\ + \frac{1}{{n!}}\frac{1}{{\gamma - \rho }}{\left( {\int\limits_\rho ^\gamma {(\omega - \rho){\omega^{np}}d\omega} } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^\gamma {(\omega - \rho)\left[ {{{(\frac{{\omega - \rho }}{{\gamma - \rho }})}^s}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + {{m(\frac{{\gamma - \omega}}{{\gamma - \rho }})}^s}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]d\omega} } \right)^{\frac{1}{q}}},\end{array} \end{equation*}$

(2.2)

Let

$\begin{equation*} \begin{array}{*{20}{l}} {{I_1} = {{\left[ {\int\limits_\rho ^\gamma {(\gamma - \omega){\omega^{np}}d\omega} } \right]}^{\frac{1}{p}}}} { = {{\left[ {\int\limits_\rho ^\gamma {(\gamma {\omega^{np}} - {\omega^{np + 1}})d\omega} } \right]}^{\frac{1}{p}}}}\\ { = {{(\gamma - \rho )}^{\frac{1}{p}}}{{\left[ {\gamma \left( {\frac{{{\gamma ^{np + 1}} - {\rho ^{np + 1}}}}{{(\gamma - \rho )(np + 1)}}} \right) - \left( {\frac{{{\gamma ^{np + 2}} - {\rho ^{np + 2}}}}{{(\gamma - \rho )(np + 2)}}} \right)} \right]}^{\frac{1}{p}}}} { = {{(\gamma - \rho )}^{\frac{1}{p}}}{{\left[ {\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )} \right]}^{\frac{1}{p}}}}, \end{array} \end{equation*}$

$\begin{equation*} \begin{array}{*{20}{l}} {{I_2} = {{\left[ {\int\limits_\rho ^\gamma {(\omega - \rho ){\omega^{np}}dt} } \right]}^{\frac{1}{p}}}} { = {{\left[ {\int\limits_\rho ^\gamma {({\omega^{np + 1}} - \rho {\omega^{np}})d\omega} } \right]}^{\frac{1}{p}}}}\\ { = {{(\gamma - \rho )}^{\frac{1}{p}}}{{\left[ {\left( {\frac{{{\gamma ^{np + 2}} - {\rho ^{np + 2}}}}{{(\gamma - \rho )(np + 2)}}} \right) - \rho \left( {\frac{{{\gamma ^{np + 1}} - {\rho ^{np + 1}}}}{{(\gamma - \rho )(np + 1)}}} \right)} \right]}^{\frac{1}{p}}}} { = {{(\gamma - \rho )}^{\frac{1}{p}}}{{\left[ {L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )} \right]}^{\frac{1}{p}}}}, \end{array} \end{equation*}$

$\begin{equation*} \begin{array}{l} {I_3} = \int\limits_\rho ^\gamma {(\gamma - \omega){{(\omega - \rho )}^s}} d\omega = (\gamma - \omega)\frac{{{{(\omega - \rho )}^{s + 1}}}}{{s + 1}}|_\rho ^\gamma + \int\limits_\rho ^\gamma {\frac{{{{(\omega - \rho )}^{s + 1}}}}{{s + 1}}} d\omega = \frac{{{{(\gamma - \rho )}^{s + 2}}}}{{(s + 1)(s + 2)}}, \end{array} \end{equation*}$

$\begin{equation*} \begin{array}{l} {I_4} = \int\limits_\rho ^\gamma {{{(\gamma - \omega)}^{s + 1}}} d\omega = \frac{{{{(\gamma - \rho )}^{s + 2}}}}{{s + 2}}, {I_5} = \int\limits_\rho ^\gamma {{{(\omega - \rho )}^{s + 1}}} d\omega = \frac{{{{(\gamma - \rho )}^{s + 2}}}}{{s + 2}}, \\ {I_6} = \int\limits_\rho ^\gamma {(\omega - \rho ){{(\gamma - \omega)}^s}} d\omega = (\omega - \rho )\frac{{{{(\gamma - \omega)}^{s + 1}}}}{{(s + 1)}}|_\rho ^\gamma + \int\limits_\rho ^\gamma {\frac{{{{(\gamma - \omega)}^{s + 1}}}}{{(s + 1)}}} d\omega = \frac{{{{(\gamma - \rho )}^{s + 2}}}}{{(s + 1)(s + 2)}}. \end{array} \end{equation*}$

Substituting integrals ${I_1}, {I_2}, {I_3}, {I_4}, {I_5}, {I_6}$ in inequality (2.2) we have,

$\begin{equation*} \begin{array}{l} \left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}} (\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}) - \int\limits_\rho ^\gamma {\chi (\omega)d\omega} } \right| \le \\ \frac{1}{{n!(\gamma - \rho )}}\left( {\begin{array}{*{20}{l}} {{{(\gamma - \rho )}^{\frac{1}{p}}}{{[\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )]}^{\frac{1}{p}}}{{\left[ {{{(\gamma - \rho )}^2}\left( {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)(s + 1)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 2)}}} \right)} \right]}^{\frac{1}{q}}}}\\ { + {{(\gamma - \rho )}^{\frac{1}{p}}}{{[L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )]}^{\frac{1}{p}}}{{\left[ {{{(\gamma - \rho )}^2}\left( {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 1)(s + 2)}}} \right)} \right]}^{\frac{1}{q}}}} \end{array}} \right) \end{array} \end{equation*}$

$\begin{equation*} \begin{array}{l} = \frac{{{{(\gamma - \rho )}^{\frac{1}{p} - 1 + \frac{2}{q}}}}}{{n!}}\left( {\begin{array}{*{20}{l}} {{{[\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )]}^{\frac{1}{p}}}{{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)(s + 1)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 2)}}} \right]}^{\frac{1}{q}}}}\\ { + {{[L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )]}^{\frac{1}{p}}}{{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 1)(s + 2)}}} \right]}^{\frac{1}{q}}}} \end{array}} \right) \end{array} \end{equation*}$

$\begin{equation*} \begin{array}{l} = \frac{1}{{n!}}{(\gamma-\rho)^{\frac{1}{q}}}\left( \begin{array}{l} {[\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )]^{\frac{1}{p}}}{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)(s + 1)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 2)}}} \right]^{\frac{1}{q}}}\\ + {[L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )]^{\frac{1}{p}}}{\left[ {\frac{{{{\left| {{\chi ^n}(\gamma )} \right|}^q}}}{{(s + 2)}} + \frac{{{{m\left| {{\chi ^n}(\rho )} \right|}^q}}}{{(s + 1)(s + 2)}}} \right]^{\frac{1}{q}}} \end{array} \right). \end{array} \end{equation*}$

which is required inequality (2.1).

For $n = 1$ inequality (2.1) becomes,

$\begin{equation} \begin{array}{*{20}{l}} {\left| {\left( {\frac{{\chi (\gamma )\gamma - \chi (\rho )\rho }}{{\gamma - \rho }}} \right) - \frac{1}{{\gamma - \rho }}\int\limits_\rho ^\gamma {\chi (\omega)d\omega} } \right| \le }\\ {{{(\gamma - \rho )}^{\frac{1}{q} - 1}}\left( {\begin{array}{*{20}{l}} {{{\left[ {\gamma L_p^p(\rho , \gamma ) - L_{p + 1}^{p + 1}(\rho , \gamma )} \right]}^{\frac{1}{p}}}{{\left[ {\frac{{{{\left| {\chi '(\gamma )} \right|}^q}}}{{(s + 1)(s + 2)}} + {{\frac{{m\left| {\chi '(\rho )} \right|}}{{(s + 2)}}}^q}} \right]}^{\frac{1}{q}}}}\\ { + {{\left[ {L_{p + 1}^{p + 1}(\rho, \gamma ) - \rho L_p^p(\rho , \gamma )} \right]}^{\frac{1}{p}}}{{\left[ {\frac{{{{m\left| {\chi '(\rho )} \right|}^q}}}{{(s + 1)(s + 2)}} + {{\frac{{\left| {\chi '(\gamma )} \right|}}{{(s + 2)}}}^q}} \right]}^{\frac{1}{q}}}} \end{array}} \right)}. \end{array} \end{equation}$

(2.3)

Remark 2.2. For $s = 1$ and $m = 1$ our resulting inequality (2.1) becomes the inequality ( $2$ ) of ^[5].

Theorem 2.3. For $n\in \mathbb{N}$ , let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be $n$ -times differentiable mapping on $U^{\circ}$ , where, $\rho, \gamma \in U^{\circ}$ , $\rho < \gamma$ , $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$ , is $(s, m)$ -convex on interval $[\rho, \gamma]$ then following inequality holds

$\begin{equation} \begin{array}{l} \left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}\left( {\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}} \right) - \int\limits_\rho ^\gamma {\chi (\omega)d\omega} } } \right| \le \\ \frac{1}{s^\frac{1}{q}{n!}}{\left( {\frac{1}{2}} \right)^{\frac{1}{p}}}{(\gamma - \rho )^{\frac{2}{p} - 1}}\\ \left( {\begin{array}{*{20}{l}} {{{\left( {\begin{array}{*{20}{l}} {\frac{{{{\left| {{\chi ^{(n)}}(\gamma )} \right|}^q}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(n)}}(\rho )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\gamma L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\gamma ^2}L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}} + }\\ {{{\left( {\begin{array}{*{20}{l}} {{{\frac{{\left| {{\chi ^{(n)}}(\gamma )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\rho L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\rho ^2}L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(n)}}(\rho )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}} \end{array}} \right) \end{array}. \end{equation}$

(2.4)

Proof. Since $|\chi^{(n)}|^{q}$ for $q > 1$ is $(s, m)$ -convex on $[\rho, \gamma]$ , by using Lemma 1.4 and Hölder-İşcan inequality (1.2), since $s \in (0, 1]$ , this fact can be used for $\omega, \rho, \gamma \in U\subseteq(0, \infty)$ ,

$\begin{equation*} \begin{array}{c} {(\omega - \rho )^s} < \frac {(\omega - \rho )}{s}, {(\gamma - \omega)^s} < \frac{(\gamma - \omega)}{s}\\ \begin{array}{l} \left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}\left( {\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}} \right) - \int\limits_\rho ^\gamma {\chi (\omega)d\omega} } } \right|\\ \le \frac{1}{{n!}}\int\limits_\rho ^\gamma {1.{\omega^n}} \left| {{\chi ^{(n)}}(\omega)} \right|d\omega, \\ \le \frac{1}{{n!}}\frac{1}{{(\gamma - \rho )}}\left( \begin{array}{l} \left[ {{{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega)d\omega} } \right)}^{\frac{1}{p}}}{{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega){\omega^{nq}}{{\left| {{\chi ^{(n)}}(\omega)} \right|}^q}d\omega} } \right)}^{\frac{1}{q}}}} \right] + \\ \left[ {{{\left( {\int\limits_\rho ^\gamma {(\omega - \rho )d\omega} } \right)}^{\frac{1}{p}}}{{\left( {\int\limits_\rho ^\gamma {(\omega - \rho ){\omega^{nq}}{{\left| {{\chi ^{(n)}}(\omega)} \right|}^q}d\omega} } \right)}^{\frac{1}{q}}}} \right] \end{array} \right), \end{array}\\ \le \frac{1}{{n!}}\frac{1}{{(\gamma - \rho )}}{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega)d\omega} } \right)^{\frac{1}{p}}}\left( {\int\limits_\rho ^\gamma {(\gamma - \omega){\omega^{nq}}\left[ {{{\left( {\frac{{\omega - \rho }}{{\gamma - \rho }}} \right)}^s}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + {{m\left( {\frac{{\gamma - \omega}}{{\gamma - \rho }}} \right)}^s}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]dt} } \right)^{\frac{1}{q}} + \\ \frac{1}{{n!}}\frac{1}{{(\gamma - \rho )}}{\left( {\int\limits_\rho ^\gamma {(\omega - \rho )dt} } \right)^{\frac{1}{p}}}\left( {\int\limits_\rho ^\gamma {(\omega - \rho ){\omega^{nq}}\left[ {{{\left( {\frac{{\omega - \rho }}{{\gamma - \rho }}} \right)}^s}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + {{m\left( {\frac{{\gamma - \omega}}{{\gamma - \rho }}} \right)}^s}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]dx} } \right)^{\frac{1}{q}}, \\ \le \frac{1}{s^{\frac{1}{q}}{n!}}\frac{1}{{(\gamma - \rho )}}{\left( {\int\limits_\rho ^\gamma {(\gamma - \omega)d\omega} } \right)^{\frac{1}{p}}}\left( {\int\limits_\rho ^\gamma {(\gamma - \omega){\omega^{nq}}\left[ {\frac{{(\omega - \rho )}}{{{{(\gamma - \rho )}^s}}}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + \frac{{m(\gamma - \omega)}}{{{{(\gamma - \rho )}^s}}}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]d\omega} } \right)^{\frac{1}{q}}\\ + \frac{1}{s^{\frac{1}{q}}{n!}}\frac{1}{{(\gamma - \rho )}}{\left( {\int\limits_\rho ^\gamma {(\omega - \rho )d\omega} } \right)^{\frac{1}{p}}}\left( {\int\limits_\rho ^\gamma {(\omega - \rho ){\omega^{nq}}\left[ {\frac{{(\omega - \rho )}}{{{{(\gamma - \rho )}^s}}}{{\left| {{\chi ^n}(\gamma )} \right|}^q} + \frac{{m(\gamma - \omega)}}{{{{(\gamma - \rho )}^s}}}{{\left| {{\chi ^n}(\rho )} \right|}^q}} \right]d\omega} } \right)^{\frac{1}{q}}, \\ \begin{array}{l} \ \ \ \ {I_1} = \int\limits_\rho ^\gamma {(\gamma - \omega )d\omega = } {\rm{ }}\frac{{{{(\gamma - \rho )}^2}}}{2} \\ \ \ \ \ {I_2} = \int\limits_\rho ^\gamma {(\gamma - \omega)(\omega - \rho ){\omega^{nq}}d\omega} = {\gamma \frac{{{\omega^{nq + 1}}}}{{nq + 1}} - \rho \gamma \frac{{{\omega^{nq + 1}}}}{{nq + 1}} - \frac{{{\omega^{nq + 3}}}}{{nq + 3}} + \frac{{\rho {\omega^{nq + 2}}}}{{nq + 2}}}|_\rho ^\gamma \\ \ \ \ \ = - \left( {\frac{{{\gamma ^{nq + 3}} - {\rho ^{nq + 3}}}}{{nq + 3}}} \right) + \rho \left( {\frac{{{\gamma ^{nq + 2}} - {\rho ^{nq + 2}}}}{{nq + 2}}} \right) + \gamma \left( {\frac{{{\gamma ^{nq + 2}} - {\rho ^{nq + 2}}}}{{nq + 2}}} \right) - \rho \gamma \left( {\frac{{{\gamma ^{nq + 1}} - {\rho ^{nq + 1}}}}{{nq + 1}}} \right)\\\ \ \ \ = (\gamma - \rho )\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right], \\ \ \ \ \ \ \ \ \ {I_3} = \int\limits_\rho ^\gamma {{{(\gamma - \omega)}^2}} {\omega^{nq}}d\omega = {{\gamma ^2}\frac{{{\omega^{nq + 1}}}}{{nq + 1}} + \frac{{{\omega^{nq + 3}}}}{{nq + 3}} - 2\gamma \frac{{{\omega^{nq + 2}}}}{{nq + 2}}} |_\rho ^\gamma \\ \ \ \ \ \ \ \ \ = \left( {\frac{{{\gamma ^{nq + 3}} - {\rho ^{nq + 3}}}}{{nq + 3}}} \right) - 2\gamma \left( {\frac{{{\gamma ^{nq + 2}} - {\rho ^{nq + 2}}}}{{nq + 2}}} \right) + {\gamma ^2}\left( {\frac{{{\gamma ^{nq + 1}} - {\rho ^{nq + 1}}}}{{nq + 1}}} \right)\\ \ \ \ \ \ \ \ \ = (\gamma - \rho )\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\gamma L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\gamma ^2}L_{nq}^{nq}(\rho , \gamma )} \right], \\ \ \ \ \ \ \ \ \ {I_4} = \int\limits_\rho ^\gamma {{{(\omega - \rho )}^2}} {\omega^{nq}}d\omega = {\frac{{{\omega^{nq + 3}}}}{{nq + 3}} + {\rho ^2}\frac{{{\omega^{nq + 1}}}}{{nq + 1}} - 2\rho \frac{{{\omega^{nq + 2}}}}{{nq + 2}}} |_\rho ^\gamma \\ \ \ \ \ \ \ \ \ = \left( {\frac{{{\gamma ^{nq + 3}} - {\rho ^{nq + 3}}}}{{nq + 3}}} \right) + {\rho ^2}\left( {\frac{{{\gamma ^{nq + 1}} - {\rho ^{nq + 1}}}}{{nq + 1}}} \right) - 2\rho \left( {\frac{{{\gamma ^{nq + 2}} - {\rho ^{nq + 2}}}}{{nq + 2}}} \right)\\ \ \ \ \ \ \ \ \ = (\gamma - \rho )\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) + {\rho ^2}L_{nq}^{nq}(\rho , \gamma ) - 2\rho L_{nq + 1}^{nq + 1}(\rho , \gamma )} \right]. \end{array} \end{array} \end{equation*}$

(2.5)

Substituting integrals ${I_1}, {I_2}, {I_3}, {I_4}, {I_5}, {I_6}$ in inequality (2.5) we have,

$\begin{equation*} \begin{array}{l} = \frac{1}{s^{\frac{1}{q}}{n!}}{\left( {\frac{1}{2}} \right)^{\frac{1}{p}}}{(\gamma - \rho )^{\frac{2}{p} - 1}}\times \\ \left( {\begin{array}{*{20}{l}} {{{\left( {\begin{array}{*{20}{l}} {\frac{{{{\left| {{\chi ^{(n)}}(\gamma )} \right|}^q}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(n)}}(\rho )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\gamma L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\gamma ^2}L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}} + }\\ {{{\left( {\begin{array}{*{20}{l}} {{{\frac{{\left| {{\chi ^{(n)}}(\gamma )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\rho L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\rho ^2}L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(n)}}(\rho )} \right|^{q}}}{{{{(\gamma - \rho )}^{s-1}}}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}} \end{array}} \right). \end{array} \end{equation*}$

For $n = 1$ , Theorem2.3 reduced to the inequality

$\begin{equation} \begin{array}{l} \left| {\frac{{\gamma \chi (\gamma ) - \rho \chi (\rho )}}{{(\gamma - \rho )}} - \frac{1}{{(\gamma - \rho )}}\int\limits_\rho ^\gamma {\chi (\omega)d\omega} } \right| \le \\ \frac{1}{{s^{\frac{1}{q}}}}{\left( {\frac{1}{2}} \right)^{\frac{1}{p}}}{(\gamma - \rho )^{\frac{2}{p} - 2}}\left( {\begin{array}{*{20}{l}} {{{\left( {\begin{array}{*{20}{l}} {\frac{{{{\left| {{\chi ^{(1)}}(\gamma )} \right|}^q}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_q^q(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(1)}}(\rho )} \right|}}{{{{(\gamma - \rho )}^{s - 1}}}}}^q}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\gamma L_{q + 1}^{q + 1}(\rho , \gamma ) + {\gamma ^2}L_q^q(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}} + }\\ {{{\left( {\begin{array}{*{20}{l}} {{{\frac{{\left| {{\chi ^{(1)}}(\gamma )} \right|}}{{{{(\gamma - \rho )}^{s - 1}}}}}^q}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\rho L_{q + 1}^{q + 1}(\rho , \gamma ) + {\rho ^2}L_q^q(\rho , \gamma )} \right] + }\\ {{{\frac{{m\left| {{\chi ^{(1)}}(\rho )} \right|}}{{{{(\gamma - \rho )}^{s - 1}}}}}^q}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_q^q(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}} \end{array}} \right). \end{array} \end{equation}$

(2.6)

Remark 2.4. For $s = 1$ and $m = 1$ our resulting inequality (2.4) becomes the inequality $(6)$ of ^[5].

2.2. Hadamard type inequality via $(s, m)$ -convexity

Theorem 2.5. If function $\chi: [0, b] \longrightarrow \mathbb{R}$ , $b > 0$ is a (s, m)-convex function in the second sense where $(s, m) \in (0, 1]^{2}$ , holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$ , then

$\begin{equation} \begin{array}{l} {2^s} {\chi \left( {\frac{{\rho + m\gamma }}{2}} \right)} \le \left[ {\frac{1}{{m\gamma - \rho }}\int\limits_\rho ^{m\gamma } {\chi (\omega )d\omega + \frac{{{m^2}}}{{m\gamma - \rho }}\int\limits_{\frac{\rho }{m}}^\gamma {\chi (l)dl} } } \right]\\ \le \frac{{\chi (\rho ) + m\chi (\gamma )}}{{s + 1}} + \frac{{\chi (\gamma ) + m\chi (\frac{\rho }{{{m^2}}})}}{{s + 1}}. \end{array} \end{equation}$

(2.7)

Proof. A function $\chi: [0, b] \longrightarrow \mathbb{R}$ , $b > 0$ is said to be a $(s, m)$ -convex function in the second sense where $s, m \in (0, 1]^{2}$ , if

$\begin{equation*} \chi \Big(\varsigma \rho+ m(1-\varsigma)\gamma\Big)\leq \varsigma^{s}\chi(\rho)+m(1-\varsigma)^{s} \chi(\gamma), \end{equation*}$

holds provided that all $\rho, \gamma \in [0, b]$ and $\varsigma \in [0, 1]$ .

Integrating w.r.t $\varsigma$ on $[0, 1]$ ,

$\begin{equation*} \begin{array}{l} \int\limits_0^1 \chi (\varsigma \rho + m(1 - \varsigma )\gamma )d\varsigma \le \int\limits_0^1 {{\varsigma ^s}\chi (\rho )d\varsigma + \int\limits_0^1 {m{{(1 - \varsigma )}^s}\chi (\gamma )d\varsigma, } } \\ \\ = {\frac{{{\varsigma ^{s + 1}}}}{{s + 1}}} |_0^1\chi (\rho ) - m\chi (\gamma ) {\frac{{{{(1 - \varsigma )}^{s + 1}}}}{{s + 1}}} |_0^1 = \frac{{\chi (\rho ) + m\chi (\gamma )}}{{s + 1}}. \\ \ \ \ \ \ \ \int\limits_0^1 \chi (\varsigma \rho + m(1 - \varsigma )\gamma ) d\varsigma \leq\frac{{\chi (\rho ) + m\chi (\gamma )}}{{s + 1}}. \end{array} \end{equation*}$

(2.8)

and

$\begin{equation*} \begin{array}{l} \chi (\varsigma \gamma + m(1 - \varsigma )\frac{\rho }{{{m^2}}}) \le {\varsigma ^s}\chi (\gamma ) + m{(1 - \varsigma )^s}\chi (\frac{\rho }{{{m^2}}}), \\ \int\limits_0^1 {\chi (\varsigma \gamma + m(1 - \varsigma )\frac{\rho }{{{m^2}}})} d\varsigma \le \frac{{\chi (\gamma ) + m\chi (\frac{\rho }{{{m^2}}})}}{{s + 1}}. \end{array} \end{equation*}$

(2.9)

As $\chi$ is $(s, m)$ -convex,

$\begin{equation*} \begin{array}{l} \chi \left( {\frac{{\rho + m\gamma }}{2}} \right) = \chi \left( {\frac{{\varsigma \rho + (1 - \varsigma )m\gamma }}{2} + m.\frac{{(1 - \varsigma )\frac{\rho }{m} + \varsigma \gamma }}{2}} \right) \\ \le {\left( {\frac{1}{2}} \right)^s}\chi (\varsigma \rho + (1 - \varsigma )\gamma m) + m{\left( {\frac{1}{2}} \right)^s}\chi (\varsigma \gamma + (1 - \varsigma )\frac{\rho }{m}), \\ \end{array} \end{equation*}$

Integrating w.r.t $\varsigma$ over $[0, 1]$ and by using (2.8) and (2.9) we get,

$\begin{equation} \begin{array}{l} {2^s}\chi \left( {\frac{{\rho + m\gamma }}{2}} \right) \le \int\limits_0^1 {\left( {\chi (\varsigma \rho + (1 - \varsigma )\gamma m} \right)d\varsigma + } m\int\limits_0^1 {\chi (\varsigma \gamma + (1 - \varsigma )\frac{\rho }{m})} d\varsigma \\ \le \frac{{\chi (\rho ) + m\chi (\gamma )}}{{s + 1}} + \frac{{\chi (\gamma ) + m\chi (\frac{\rho }{{{m^2}}})}}{{s + 1}}. \end{array} \end{equation}$

(2.10)

Substituting in first integral,

${ { \varsigma \rho + (1 - \varsigma)\gamma m}} = \omega$ ,

$\begin{equation} \begin{array}{l} \int\limits_0^1 {\chi (\varsigma \rho + (1 - \varsigma )m\gamma )d\varsigma = \frac{1}{{\gamma m - \rho }}} \int\limits_\rho ^{\gamma m} {\chi (\omega)d\omega}. \\ \end{array} \end{equation}$

(2.11)

Substituting in the second integral,

$\varsigma \gamma + (1 - \varsigma)\frac{\rho }{m} = l$ ,

$\begin{equation} \begin{array}{l} \int\limits_0^1 {\chi (\varsigma \gamma + (1 - \varsigma )\frac{\rho }{m} )d\varsigma = \frac{m}{{\gamma m - \rho }}} \int\limits_{\frac{\rho }{m}}^\gamma {\chi (l)dl}, {\rm{ }} \end{array} \end{equation}$

(2.12)

Using (2.11) and (2.12) in (2.10) required inequality (2.7) obtained.

Remark 2.6. For $s, m = 1$ inequality (2.7) becomes classical Hadamard inequality for convex functions.

2.3. Application of Hadamrd type inequality via $(s, m)$ -concavity

Theorem 2.7. For $n\in \mathbb{N}$ , let $\chi:U\subseteq(0, \infty)\rightarrow \mathbb{R}$ be n-times differentiable mapping on $U^{\circ}$ , where, $\rho, \gamma \in U^{\circ}$ , $\rho < \gamma$ and $\chi^{(n)} \in L[\rho, \gamma]$ and $|\chi^{(n)} |^{q}$ for $q > 1$ is (s, m)-concave on interval $[\rho, m\gamma]$ , then

(2.13)

Proof. $|\chi^{(n)} |^{q}$ for $q > 1$ is $(s, m)$ -concave then by using Theorem 2.5 we have,

$\begin{equation*} \begin{array}{l} \frac{{{{\left| {{\chi ^{(n)}}(\rho )} \right|}^q} + m{{\left| {{\chi ^{(n)}}(\gamma )} \right|}^q}}}{{s + 1}} + \frac{{{{\left| {{\chi ^{(n)}}(\gamma )} \right|}^q} + m{{\left| {{\chi ^{(n)}}(\frac{\rho }{{{m^2}}})} \right|}^q}}}{{s + 1}} - \frac{{{m^2}}}{{(m\gamma - \rho )}}\int\limits_{\frac{\rho }{m}}^\gamma {{{\left| {{\chi ^{(n)}}(l)} \right|}^q}dl} \\ \le \frac{1}{{(m\gamma - \rho )}}\int\limits_\rho ^{m\gamma } {{{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega } \le {2^s}{\left| {{\chi ^{(n)}}\left( {\frac{{\rho + m\gamma }}{2}} \right)} \right|^q}, \end{array} \end{equation*}$

$\begin{equation*} \int\limits_\rho ^{m\gamma } {{{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega } \le {2^s}(m\gamma - \rho ){\left| {{\chi ^{(n)}}\left( {\frac{{\rho + m\gamma }}{2}} \right)} \right|^q}, \end{equation*}$

$\begin{equation*} \frac{1}{{(m\gamma - \rho )}}\int\limits_\rho ^{\gamma m} {(\gamma - \omega ){{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega \le } \int\limits_\rho ^{\gamma m} {{{\left| {{\chi ^{(n)}}(\omega )} \right|}^q}d\omega }\le {2^s}(m\gamma - \rho ){\left| {{\chi ^{(n)}}\left( {\frac{{\rho + m\gamma }}{2}} \right)} \right|^q}, \end{equation*}$

Using Lemma 1.4 and Hölder-Îşcan inequality (1.2),

$\begin{equation*} \begin{array}{l}\left| {\sum\limits_{\nu = 0}^{n - 1} {{{( - 1)}^\nu}\left( {\frac{{{\chi ^{(\nu)}}(\gamma ){\gamma ^{\nu + 1}} - {\chi ^{(\nu)}}(\rho ){\rho ^{\nu + 1}}}}{{(\nu + 1)!}}} \right) - \int\limits_\rho ^{\gamma m} {\chi (\omega)d\omega} } } \right| \le \frac{1}{{n!}}\int\limits_\rho ^{\gamma m} {{\omega^n}} \left| {{\chi ^{(n)}}(\omega)} \right|d\omega, \\ \ \ \ \ \ \ \le \frac{1}{{n!}}\frac{1}{{\gamma - \rho }}\left\{ \begin{array}{l} {\left( {\int\limits_\rho ^{\gamma m} {(\gamma - \omega ){\omega ^{np}}d\omega } } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^{m\gamma } {(\gamma - \omega ){{\left| {{\chi ^n}(\omega )} \right|}^q}d\omega } } \right)^{\frac{1}{q}}} + \\ {\left( {\int\limits_\rho ^{\gamma m} {(\omega - \rho ){\omega ^{np}}d\omega } } \right)^{\frac{1}{p}}}{\left( {\int\limits_\rho ^{m\gamma } {(\omega - \rho ){{\left| {{\chi ^n}(\omega )} \right|}^q}d\omega } } \right)^{\frac{1}{q}}} \end{array} \right\}, \\ \ \ \ \ \ \ \le \frac{1}{{n!}}\frac{1}{{\gamma - \rho }}\left( {\begin{array}{*{20}{l}} \begin{array}{l} {\left( {\int\limits_\rho ^{\gamma m} {(\gamma - \omega ){\omega ^{np}}d\omega } } \right)^{\frac{1}{p}}}{\left( {{2^s}{{(m\gamma - \rho )}^2}{{\left| {{\chi ^{(n)}}\left( {\frac{{\rho + m\gamma }}{2}} \right)} \right|}^q}} \right)^{\frac{1}{q}}}+ \\ {\left( {\int\limits_\rho ^{\gamma m} {(\omega - \rho ){\omega^{np}}d\omega} } \right)^{\frac{1}{p}}} {\left( {{2^s}{{(m\gamma - \rho )}^2}{{\left| {{\chi ^{(n)}}\left( {\frac{{\rho + m\gamma }}{2}} \right)} \right|}^q}} \right)^{\frac{1}{q}}} \end{array} \end{array}} \right), \\ \ \ \ \ \ \ \ \ \ \ \ \ {I_1} = \left( {\int\limits_\rho ^{\gamma m}{(\gamma - \omega){\omega^{np}}d\omega} } \right)^{\frac{1}{p}} = \left( {\gamma \frac{{{\omega^{np + 1}}}}{{np + 1}}| _\rho ^{\gamma m}- \frac{{{\omega^{np + 2}}}}{{np + 2}}}| _\rho ^{\gamma m} \right)^{\frac{1}{p}}\\ \ \ \ \ \ \ \ \ \ \ \ \ = {(m\gamma-\rho)^{\frac{1}{p}}}\left(\gamma L_{np}^{np}(\rho , m\gamma ) - L_{np + 1}^{np + 1}(\rho , m\gamma )\right)^{\frac{1}{p}}, \\ \ \ \ \ \ \ \ \ \ \ \ \ {I_2} = {\left( {\int\limits_\rho ^{\gamma m} {(\omega - \rho ){\omega^{np}}d\omega} } \right)^{\frac{1}{p}}} = {\left( {\frac{{{\omega^{np + 2}}}}{{np + 2}}| _\rho^{\gamma m}- \rho \frac{{{\omega^{np + 1}}}}{{np + 1}}}|_\rho^{\gamma m} \right)^{\frac{1}{p}}}\\\ \ \ \ \ \ \ \ \ \ \ \ = {(m\gamma-\rho)}^{\frac{1}{p}}\left( L_{np + 1}^{np + 1}(\rho , m\gamma ) - \rho L_{np}^{np}(\rho , m\gamma )\right)^{\frac{1}{p}}. \end{array} \end{equation*}$

(2.14)

Substituting integrals ${I_1}, {I_2}$ in inequality (2.14) required inequality (2.13) is obtained.

For $n = 1$ inequality (2.13) becomes,

$\begin{equation} \begin{array}{l} \left| {\frac{{\chi (\gamma )\gamma - \rho \chi (\rho )}}{{(\gamma - \rho )}} - \frac{1}{{(\gamma - \rho )}}\int\limits_\rho ^{\gamma m} {\chi (\omega)d\omega} } \right| \le \\ \frac{{{2^{\frac{s}{q}}}{{(m\gamma - \rho )}^{\frac{1}{q}}}\left| {{\chi ^{(1)}}\left( {\frac{{\rho + \gamma }}{2}} \right)} \right|}}{1!}\left( \begin{array}{l} {\left( {\gamma L_p^p(\rho , m\gamma ) - L_{p + 1}^{p + 1}(\rho , m\gamma )} \right)^{\frac{1}{p}}}\\ + {\left( {L_{p + 1}^{p + 1}(\rho , m\gamma ) - \rho L_p^p(\rho , m\gamma )} \right)^{\frac{1}{p}}} \end{array} \right). \end{array} \end{equation}$

(2.15)

Remark 2.8. For $s = 1$ and $m = 1$ our resulting inequality becomes the inequality obtained in Theorem $4$ of ^[5].

2.4. Applications of newly obtained inequalities to means

Proposition 2.9. Let $\rho, \gamma \in (0, \infty)$ , where $\rho < \gamma$ , $q > 1$ , $n, i \in \mathbb{N}$ with $i\geq n$ ,

$\begin{equation} \begin{array}{l} \left| {L_i^i(\rho , \gamma )\left[ {(i + 1)\sum\nolimits_{\nu = 0}^{n - 1} {\frac{{{{( - 1)}^\nu}P(i, \nu)}}{{(\nu + 1)!}} - 1} } \right]} \right| \le \frac{1}{{n!}}{(\gamma - \rho )^{\frac{1}{q} - 1}}\\ \times \left( \begin{array}{l} {\left[ {\gamma L_{np}^{np}(\rho , \gamma ) - L_{np + 1}^{np + 1}(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left( {\frac{{{\gamma ^{(i - n)q}}}}{{(s + 1)(s + 2)}} + \frac{{{m\rho ^{(i - n)q}}}}{{(s + 2)}}} \right)^{\frac{1}{q}}}\\ + {\left[ {L_{np + 1}^{np + 1}(\rho , \gamma ) - \rho L_{np}^{np}(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left( {\frac{{{m\rho ^{(i - n)q}}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^{(i - n)q}}}}{{(s + 2)}}} \right)^{\frac{1}{q}}} \end{array} \right), \end{array} \end{equation}$

(2.16)

where

$\begin{equation*} P(i, n) = \left\{ \begin{array}{l} i(i - 1)...(i - n + 1), i > n\\ n!, \quad i = n\\ 1, \quad n = 0 \end{array} \right\}. \end{equation*}$

Proof. Let

$\begin{equation*} \chi (\omega) = {\omega^i}, {\left| {{\chi ^{(n)}}(\omega)} \right|^q} = {\left| {P(i, n){\omega^{i - n}}} \right|^q}\\ \end{equation*}$

Let

$\begin{equation*} g(\varsigma) = {\left| {P(i, n)(\varsigma \rho + m(1 - \varsigma )\gamma } \right|^{(i - n)q}} - {\left| {P(i, n){\varsigma ^s}\rho } \right|^{(i - n)q}} - {\left| m{P(i, n){{(1 - \varsigma )}^s}\gamma } \right|^{(i - n)q}}, \end{equation*}$

$\begin{equation*} \begin{array}{l} {g^{''}}(\varsigma ) = P(i, n)((i - n)q)((i - n)q - 1){(\varsigma \rho + m(1 - \varsigma )\gamma )^{(i - n)q - 2}}{(\rho - m\gamma )^2} - \\ s(s - 1){\varsigma ^{s - 2}}P(i, n){\rho ^{(i - n)q}} - ms(s - 1){(1 - \varsigma )^{s - 2}}P(i, n){\gamma ^{(i - n)q}}, \end{array} \end{equation*}$

${g^{''}}(\varsigma) \ge 0$ means g is convex and $g(1) = g(0) = 0$ , which omplies $g\leq 0$ , hence

$\begin{equation*} {\left| {P(i, n)(\varsigma \rho + m(1 - \varsigma )\gamma) } \right|^{(i - n)q}} \le {\left| {P(i, n){\varsigma ^s}\rho } \right|^{(i - n)q}} + \left| m{P(i, n){{(1 - \varsigma )}^s}\gamma) } \right|^{(i-n)q}. \end{equation*}$

By using Theorem 2.1 for $|\chi^{n}(\omega)|^{q}$ which is $(s, m)$ –convex for $s, m \in (0, 1]^{2}$ inequality (2.16) obtained.

Remark 2.10. For $s, m = 1$ inequality (2.16) becomes inequality $(3)$ of ^[5].

Example 2.11. Taking $i = 2$ , $n = 1$ , $p = q = 2$ in Proposition 2.9, the following is valid:

$\begin{equation*} \begin{array}{l} 2A({\rho ^2}, {\gamma ^2}) + {G^2}(\rho , \gamma ) \le (\frac{3}{{2\sqrt 6 }})\left( \begin{array}{l} {\left[ {A(3{\rho ^2}, {\gamma ^2}) + {G^2}(\rho , \gamma )} \right]^{\frac{1}{2}}}{\left( {\frac{{{\gamma ^2}}}{{(s + 1)(s + 2)}} + \frac{{{m\rho ^2}}}{{(s + 2)}}} \right)^{\frac{1}{2}}}\\ + {\left[ {A({\rho ^2}, 3{\gamma ^2}) + {G^2}(\rho , \gamma )} \right]^{\frac{1}{2}}}{\left( {\frac{{{m\rho ^2}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^2}}}{{(s + 2)}}} \right)^{\frac{1}{2}}} \end{array} \right), \end{array} \end{equation*}$

where $A$ and $G$ are classical arithmetic and geometric means, respectively.

Proposition 2.12. Let $\rho, \gamma \in (0, \infty)$ , with, $\rho < \gamma$ , $q > 1$ and $n \in \mathbb{N}$ ,

$\begin{equation} \begin{array}{l} 1 \le {(\gamma - \rho )^{\frac{1}{q} - 1}}\left( \begin{array}{l} {\left[ {\gamma L_p^p(\rho , \gamma ) - L_{p + 1}^{p + 1}(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left[ {(\frac{{{\gamma ^{ - q}}}}{{(s + 1)(s + 2)}} + \frac{{{m\rho ^{ - q}}}}{{(s + 2)}})} \right]^{\frac{1}{q}}} + \\ {\left[ { L_{p + 1}^{p + 1}(\rho , \gamma ) -\rho L_p^p(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left[ {(\frac{{{m\rho ^{ - q}}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^{ - q}}}}{{(s + 2)}})} \right]^{\frac{1}{q}}} \end{array} \right), \end{array} \end{equation}$

(2.17)

where $L$ is classical logarithmic mean.

Proof.

$\begin{equation*} \chi (\omega) = \ln \omega, {\left| {{\chi ^{(1)}}(\omega)} \right|^q} = {\left| {{\omega^{ - 1}}} \right|^q} \end{equation*}$

Let

$\begin{equation*} g(\varsigma) = {\left| {(\varsigma \rho + m(1 - \varsigma )\gamma } \right|^{ - q}} - {\left| {{\varsigma ^s}\rho } \right|^{ - q}} - {\left| {{{m(1 - \varsigma )}^s}\gamma } \right|^{ - q}} \end{equation*}$

$\begin{equation*} \begin{array}{l} {g^{''}}(\varsigma) = ( - q)( - q - 1){(\varsigma \rho + m(1 - \varsigma )\gamma )^{ - q - 2}}{(\rho - m\gamma )^2}\\ - s(s - 1){\varsigma ^{s - 2}}{\rho ^{ - q}} - ms(s - 1){(1 - \varsigma )^{s - 2}}{\gamma ^{-q}}, \end{array} \end{equation*}$

${g^{''}}(\varsigma) \ge 0$ means $g$ is convex and $g(1) = g(0) = 0$ which implies $g\leq 0$ as

$\begin{equation*} {\left| {(\varsigma \rho +m (1 - \varsigma )\gamma } \right|^{ - q}} \le {\left| {{\varsigma ^s}\rho } \right|^{ - q}}\\ + {\left| {{{m(1 - \varsigma )}^s}\gamma } \right|^{ - q}}. \end{equation*}$

So ${\left| {{\chi ^{(1)}}(\omega)} \right|^q}$ is $(s, m)$ -convex. Then by using inequality (2.3) required inequality (2.17) obtained.

Remark 2.13. For $s, m = 1$ inequality (2.17) becomes $(4)$ of ^[5].

Example 2.14. For $n = 1$ and $p = q = 2$ , Proposition 2.12 gives:

$\begin{equation*} \label{45} 1 \le \frac{1}{{\sqrt 6 }}\left( {\begin{array}{*{20}{l}} {{{\left[ {A(3{\rho ^2}, {\gamma ^2}) + {G^2}(\rho , \gamma )} \right]}^{\frac{1}{p}}}{{\left[ {(\frac{{{\gamma ^{ - 2}}}}{{(s + 1)(s + 2)}} + \frac{{m{\rho ^{ - 2}}}}{{(s + 2)}})} \right]}^{\frac{1}{2}}} + }\\ {{{\left[ {A({\rho ^2}, 3{\gamma ^2}) + {G^2}(\rho , \gamma )} \right]}^{\frac{1}{p}}}{{\left[ {(\frac{{m{\rho ^{ - 2}}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^{ - 2}}}}{{(s + 2)}})} \right]}^{\frac{1}{2}}}} \end{array}} \right). \end{equation*}$

Proposition 2.15. Let $\rho, \gamma \in (0, \infty)$ , $\rho < \gamma$ , $q > 1$ , $i \in (- \infty, 0] \cup [1, \infty)\backslash\{ - 2q, - q\}$

then

$\begin{equation} L_{\frac{i}{q} + 1}^{\frac{i}{q} + 1}(\rho , \gamma ) \le {(\gamma - \rho )^{\frac{1}{q} - 1}}\left( \begin{array}{l} {\left[ {\gamma L_p^p(\rho , \gamma ) - L_{p + 1}^{p + 1}(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left[ {(\frac{{{\gamma ^i}}}{{(s + 1)(s + 2)}} + \frac{{m{\rho ^i}}}{{(s + 2)}})} \right]^{\frac{1}{q}}}\\ + {\left[ {L_{p + 1}^{p + 1}(\rho , \gamma ) - \rho L_p^p(\rho , \gamma )} \right]^{\frac{1}{p}}}{\left[ {(\frac{{m{\rho ^i}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^i}}}{{(s + 2)}})} \right]^{\frac{1}{q}}} \end{array} \right). \end{equation}$

(2.18)

Proof.

$\begin{equation*} \chi (t) = \frac{q}{{i + q}}{\omega^{\frac{i}{q} + 1}}, |\chi^{'} (\omega)|^{q} = \omega^{i} \end{equation*}$

Let

$\begin{equation*} g(\varsigma) = {\left| {(\varsigma \rho + m(1 - \varsigma )\gamma } \right|^i} - {\left| {{\varsigma ^s}\rho } \right|^i} - {\left| {{{m(1 - \varsigma )}^s}\gamma } \right|^i}, \end{equation*}$

$\begin{equation*} {g^{''}}(\varsigma) = (i)(i - 1){(\varsigma \rho +m (1 - \varsigma )\gamma )^{i - 2}}{(\rho - m\gamma )^2} - s(s - 1){\varsigma ^{s - 2}}{\rho ^i} - ms(s - 1){(1 - \varsigma )^{s - 2}}{\gamma ^i}, \end{equation*}$

$g^{''}(\varsigma) \geq 0$ and $g(1) = g(0)$ so $g\leq0$ and $|\chi^{'} (\omega)|^{q}$ is $(s, m)$ -convex, by using inequality (2.3) we have (2.18).

Remark 2.16. For $s, m = 1$ inequality (2.18) becomes $(5)$ of ^[5].

Example 2.17. For $i = 2$ and $p = q = 2$ Proposition 2.15 reduced to

$\begin{equation} \begin{array}{l} 2A({\rho ^2}, {\gamma ^2}) + {G^2}(\rho , \gamma ) \le \\ {(\frac{3}{\sqrt{6}})}\left( \begin{array}{l} {\left[ {A(3{\rho ^2}, {\gamma ^2}) + {G^2}(\rho , \gamma )} \right]^{\frac{1}{2}}}{\left[ {(\frac{{{\gamma ^2}}}{{(s + 1)(s + 2)}} + \frac{{{m\rho ^2}}}{{(s + 2)}})} \right]^{\frac{1}{2}}} + \\ {\left[ {A({\rho ^2}, 3{\gamma ^2}) + {G^2}(\rho , \gamma )} \right]^{\frac{1}{2}}}{\left[ {(\frac{{{m\rho ^2}}}{{(s + 1)(s + 2)}} + \frac{{{\gamma ^2}}}{{(s + 2)}})} \right]^{\frac{1}{2}}} \end{array} \right). \end{array} \end{equation}$

(2.19)

Proposition 2.18. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ , $q > 1$ and $n \in \mathbb{N}$ then we have

$\begin{equation} \times \begin{array}{*{20}{l}} {\left| {L_i^i(\rho , \gamma )\left[ {\sum\limits_{\nu = 0}^{n - 1} {\frac{{{{( - 1)}^\nu}P(i, \nu)}}{{(\nu + 1)!}} - 1} } \right]} \right| \le \frac{{P(i, n)}}{s^{\frac{1}{q}}{n!}}{{\left( {\frac{1}{2}} \right)}^{\frac{1}{p}}}{{(\gamma - \rho )}^{\frac{2}{p} - 1}}}\\ {{{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^{(i - n)q}}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^{(i - n)q}}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\gamma L_{nq + 1}^{nq + 1}(\kappa , \mu ) + {\mu ^2}L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}}\\ { + \frac{{P(i, n)}}{s^{\frac{1}{q}}{n!}}{{\left( {\frac{1}{2}} \right)}^{\frac{1}{p}}}{{(\gamma - \rho )}^{\frac{2}{p} - 1}}{{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^{(i - n)q}}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ {L_{nq + 2}^{nq + 2}(\rho , \gamma ) - 2\rho L_{nq + 1}^{nq + 1}(\rho , \gamma ) + {\rho ^2}L_{nq}^{nq}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^{(i - n)q}}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ { - L_{nq + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{nq + 1}^{nq + 1}(\rho , \gamma ) - \rho \gamma L_{nq}^{nq}(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}}. \end{array} \end{equation}$

(2.20)

Proof. Let,

$\begin{equation*} \chi (\omega) = {\omega^i}\\, {\left| {{\chi ^{(n)}}(\omega)} \right|^q} = {\left[ {P(i, n){\omega^{i - n}}} \right]^q} \end{equation*}$

As $|\chi^{n}(\omega)|^{q}$ is $(s, m)$ -convex on $(0, \infty)$ , therefore by using Theorem 2.3 required inequality (2.20) is obtained.

Remark 2.19. For $s, m = 1$ inequality (2.20) becomes inequality obtained in Proposition $4$ of ^[5].

Proposition 2.20. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ $q > 1$ and $n \in \mathbb{N}$ then we have,

$\begin{equation} 1 \le \frac{{{{(\gamma - \rho )}^{\frac{2}{p} - 2}}}}{{{s^{\frac{1}{q}}.2^{\frac{1}{p}}}}}\left( \begin{array}{l} {\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^{ - q}}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_{q}^{q}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^{ - q}}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\gamma L_{q + 1}^{q + 1}(\rho , \gamma ) + {\gamma ^2}L_{q}^{q}(\rho , \gamma )} \right]} \end{array}} \right)^{\frac{1}{q}}}\\ +{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^{ - q}}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\rho L_{q + 1}^{q + 1}(\rho , \gamma ) + {\rho ^2}L_{q}^{q}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^{ - q}}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_{q}^{q}(\rho , \gamma )} \right]} \end{array}} \right)^{\frac{1}{q}}} \end{array} \right), \end{equation}$

(2.21)

Proof.

$\begin{equation*} \chi (\omega) = \ln \omega, {\left| {{\chi ^{(1)}}(\omega)} \right|^q} = {\left[ {{\omega^{ - 1}}} \right]^q} \end{equation*}$

As ${\left| {{\chi ^{(1)}}(\omega)} \right|^q}$ is $(s, m)$ –convex, therefore by using inequality (2.6) required (2.21) obtained.

Remark 2.21. For $s, m = 1$ inequality (2.21) becomes inequality obtained in Proposition $5$ of ^[5].

Proposition 2.22. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ $q > 1$ and $i \in (-\infty, 0]\backslash\{-2q, q\}$ , then

$\begin{equation} L_{\frac{i}{q} + 1}^{\frac{i}{q} + 1}(\rho , \gamma ) \le \frac{{{{(\gamma - \rho )}^{\frac{2}{p} - 2}}}}{{{s^{\frac{1}{q}}.2^{\frac{1}{p}}}}}\left( \begin{array}{l} {\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^i}}}{{{{(\gamma - \rho )}^{s-1}{}}}}\left[ { - L_{q + 2}^{nq + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_{q}^{q}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^i}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\gamma L_{q + 1}^{q + 1}(\rho , \gamma ) + {\gamma ^2}L_{q}^{q}(\rho , \gamma )} \right]} \end{array}} \right)^{\frac{1}{q}}}\\ +{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^i}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\rho L_{q + 1}^{q + 1}(\rho , \gamma ) + {\rho ^2}L_{q}^{q}(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^m}}}{{{{(\gamma - \rho )}^{s-1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_{q}^{q}(\rho , \gamma )} \right]} \end{array}} \right)^{\frac{1}{q}}} \end{array} \right). \end{equation}$

(2.22)

Proof.

$\begin{equation*} \chi (\omega) = \frac{q}{{i + q}}{\omega^{\frac{i}{q} + 1}} |\chi^{'} (\omega)|^{q} = \omega^{i} \end{equation*}$

$|\chi^{'} (w)|^{q}$ is $(s, m)$ -convex by using inequality (2.6) required (2.22) obtained.

For $i = 1$ inequality (2.22) becomes,

$\begin{equation} L_{\frac{1}{q} + 1}^{\frac{1}{q} + 1}(\rho , \gamma ) \le \frac{{{{(\gamma - \rho )}^{\frac{2}{p} - 2}}}}{{{s^{\frac{1}{q}}.2^{\frac{1}{p}}}}}\left( {\begin{array}{*{20}{l}} {{{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma^1}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_q^q(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^1}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\gamma L_{q + 1}^{q + 1}(\rho , \gamma ) + {\gamma ^2}L_q^q(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}}\\ +{{{\left( {\begin{array}{*{20}{l}} {\frac{{{\gamma ^1}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ {L_{q + 2}^{q + 2}(\rho , \gamma ) - 2\rho L_{q + 1}^{q + 1}(\rho , \gamma ) + {\rho ^2}L_q^q(\rho , \gamma )} \right] + }\\ {\frac{{{m\rho ^1}}}{{{{(\gamma - \rho )}^{s - 1}}}}\left[ { - L_{q + 2}^{q + 2}(\rho , \gamma ) + (\rho + \gamma )L_{q + 1}^{q + 1}(\rho , \gamma ) - \rho \gamma L_q^q(\rho , \gamma )} \right]} \end{array}} \right)}^{\frac{1}{q}}}} \end{array}} \right). \end{equation}$

(2.23)

Remark 2.23. For $s, m = 1$ inequality (2.22) becomes inequality obtained in Proposition $6$ of ^[5].

Proposition 2.24. Let $\rho, \gamma \in (0, \infty)$ with $\rho < \gamma$ , $q > 1$ and $i \in [0, 1]$ we have,

$\begin{equation} \begin{array}{l} L_{\frac{i}{q} + 1}^{\frac{i}{q} + 1}(\rho , \gamma ) \le \\ \frac{{{2^{\frac{s}{q}}}{{(m\gamma - \rho )}^{\frac{1}{q}}}}}{{1!}} {A^{\frac{i}{q}}}(\rho , \gamma )\left( \begin{array}{l} {\left( {\gamma L_p^p(\rho , m\gamma ) - L_{p + 1}^{p + 1}(\rho , m\gamma )} \right)^{\frac{1}{p}}}\\ + {\left( {L_{p + 1}^{p + 1}(\rho , m\gamma ) - \rho L_p^p(\rho , m\gamma )} \right)^{\frac{1}{p}}} \end{array} \right). \end{array} \end{equation}$

(2.24)

Proof.

$\begin{equation*} \chi (\omega) = \frac{q}{{i + q}}{\omega^{\frac{i}{q} + 1}}, |\chi^{'} (\omega)|^{q} = \omega^{i}. \end{equation*}$

As $|\chi^{'} (\omega)|^{q}$ is $(s, m)$ -concave by using inequality (2.15) we obtain required inequality (2.24).

Remark 2.25. For $s, m = 1$ inequality (2.24) becomes the inequality obtained in Proposition $9$ of ^[5].

3. Conclusions

In this paper, Hölder-Isçan inequality is utilized to prove Hermite-Hadamard type inequalities for $n$ -times differentiable $(s, m)$ -convex functions. The method is adequate and provide many generalizations of existing results as shown in remarks. Moreover, many other inequalities can be generalized for other types of convex functions.

Acknowledgments

This research received funding support from the NSRF via the Program Management Unit for Human Resources & Institutional Development, Research and Innovation, (grant number B05F650018)

Conflict of interest

The authors declare no conflict of interest.

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AIMS Molecular Science

The functional roles of T-cadherin in mammalian biology

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Abstract

1. Introduction

2. Main results

2.1. Modified inequalities for $n$ -times differentiable convex functions

2.2. Hadamard type inequality via $(s, m)$ -convexity

2.3. Application of Hadamrd type inequality via $(s, m)$ -concavity

2.4. Applications of newly obtained inequalities to means

3. Conclusions

Acknowledgments

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AIMS Molecular Science

The functional roles of T-cadherin in mammalian biology

Related Papers:

Abstract

1. Introduction

2. Main results

2.1. Modified inequalities for n n -times differentiable convex functions

2.2. Hadamard type inequality via (s,m) (s, m) -convexity

2.3. Application of Hadamrd type inequality via (s,m) (s, m) -concavity

2.4. Applications of newly obtained inequalities to means

3. Conclusions

Acknowledgments

Conflict of interest

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2.1. Modified inequalities for $n$ -times differentiable convex functions

2.2. Hadamard type inequality via $(s, m)$ -convexity

2.3. Application of Hadamrd type inequality via $(s, m)$ -concavity