New Generalized Class of Convex Functions and Some Related Integral Inequalities

Abstract

There is a strong correlation between convexity and symmetry concepts. In this study, we investigated the new generic class of functions called the $(n,m)$–generalized convex and studied its basic algebraic properties. The Hermite–Hadamard inequality for the $(n,m)$–generalized convex function, for the products of two functions and of this type, were proven. Moreover, this class of functions was applied to several known identities; midpoint-type inequalities of Ostrowski and Simpson were derived. Our results are extensions of many previous contributions related to integral inequalities via different convexities.

1. Introduction and Preliminaries

The twenty-first century began with the introduction and establishment of new tools used to solve linear and nonlinear differential and difference equations. In terms of the convexity theory, one important development involves defining a new class of convex functions, which is then tested on the well-known inequalities. “As it is known, inequalities aim to develop different mathematical methods. Nowadays, we need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. In recent years, especially over the past two decades, several authors have been engaged in the study of inequalities, including various function classes (symmetric or asymmetric)”, see [1]. Moreover, the modern convexity theory has motivated researchers to propose a new generalized class of convex functions and to investigate their special models, which could effectively be used in different fields, in particular, agriculture, medicine, reliability engineering, demography, actuarial study, survival analysis, and others. Kasamsetty et al. in [2] defined a new class of convex functions used to delay modeling and established an application to the transistor sizing problem. Awan et al. in [3] obtained new classes of convex functions and inequalities. Hudzik and Maligranda in [4] investigated the class of s-convex functions. Eftekhari in [5] derived new results using $(s,m)$–convexity in the second sense. Kadakal and İşcan in [6] established related inequalities via the exponential type convexity. Agarwal and Choi in [7] used fractional operators and found their image formulas. Rekhviashvili et al. in [8] described damped vibrations via a fractional oscillator model.

In much of the literature, we can see various Hermite–Hadamard ($\mathtt{HH}$) inequality types, in which one of the known classes of convex functions is utilized (e.g., [9,10,11]). Moreover, some generalizations of the $\mathtt{HH}$ integral inequalities, such as $\mathtt{HH}$–Fejér, AB $\mathtt{HH}$, midpoint $\mathtt{HH}$, mid-end-point $\mathtt{HH}$, conformable $\mathtt{HH}$, and $\mathtt{HH}$–Mercer integral inequalities are found (e.g., [12,13,14]). In addition, different integral inequalities using those convexities are investigated. Ujević in [15] obtained sharp inequalities for Simpson and Ostrowski types. Liu et al. in [16], using the $MT$–convexity class derived Ostrowski fractional inequalities. Kaijser et al. in [17] established Hardy-type inequalities via convexity. Rashid et al. in [18], using generalized k–fractional integrals, found Grüss inequalities. For more recent published papers on $\mathtt{HH}$, see [19,20].

Let us review some fundamental and preliminary results on convexity and inequality.

Definition 1.

Function $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ is called convex, if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \varrho \mathsf{\Theta }\left({\chi }_1\right)+(1-\varrho )\mathsf{\Theta }\left({\chi }_2\right),$$

(1)

holds for all ${\chi }_1,{\chi }_2\in \tau$ (τ is an interval with real numbers and $\mathrm{R}$ is the set of real numbers) and $\varrho \in [0,1]$. Moreover, Θ is concave if $(-\mathsf{\Theta })$ is convex.

Definition 2

([4]). Let $s\in (0,1]$ be a real number. A function $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ is called s-convex (in the second sense), if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le {\varrho }^s\mathsf{\Theta }\left({\chi }_1\right)+{(1-\varrho )}^s\mathsf{\Theta }\left({\chi }_2\right),$$

(2)

holds for all ${\chi }_1,{\chi }_2\in \tau$, and $\varrho \in [0,1]$.

Definition 3

([21]). Let $\tau ,\mathcal{J}$ be intervals in $\mathrm{R},(0,1)\subseteq \mathcal{J}$ and let $h:\mathcal{J}\to \mathrm{R}$ be a nonnegative function, and $h\ne 0$. A nonnegative function $\mathsf{\Theta }:\tau \to \mathrm{R}$ is called h-convex, if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le h\left(\varrho \right)\mathsf{\Theta }\left({\chi }_1\right)+h(1-\varrho )\mathsf{\Theta }\left({\chi }_2\right),$$

(3)

holds for all ${\chi }_1,{\chi }_2\in \tau ,\varrho \in (0,1).$

Toply et al. [22] introduced the following class of convex functions:

Definition 4.

Let $n\in \mathbb{N}.$ A function $\mathsf{\Theta }:\tau \to \mathrm{R}$ is called n–polynomial convex, if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \frac{1}{n}\sum _{{\ell }_1=1}^n\left[1-{(1-\varrho )}^{{\ell }_1}\right]\mathsf{\Theta }\left({\chi }_1\right)+\frac{1}{n}\sum _{{\ell }_1=1}^n\left[1-{\varrho }^{{\ell }_1}\right]\mathsf{\Theta }\left({\chi }_2\right),$$

(4)

holds for every ${\chi }_1,{\chi }_2\in \tau ,$ and $\varrho \in [0,1]$.

Recently, Rashid et al. [23] defined the following class of convex functions:

Definition 5

([23]). Assume that $s\in [0,1]$ and $n\in \mathbb{N}.$ A function $\mathsf{\Theta }:\tau \to \mathrm{R}$ is said to be n–polynomial s–type convex, if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \frac{1}{n}\sum _{{\ell }_1=1}^n\left[1-{\left(s(1-\varrho )\right)}^{{\ell }_1}\right]\mathsf{\Theta }\left({\chi }_1\right)+\frac{1}{n}\sum _{{\ell }_1=1}^n\left[1-{\left(s\varrho \right)}^{{\ell }_1}\right]\mathsf{\Theta }\left({\chi }_2\right),$$

(5)

holds for every ${\chi }_1,{\chi }_2\in \tau ,$ and $\varrho \in [0,1]$.

The following double inequality, namely the $\mathtt{HH}$ inequality, is remarkable, and it played an important role in the analysis.

Theorem 1

($\mathtt{HH}$ inequality [24]). Let $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ be a convex function on τ for ${\chi }_1,{\chi }_2\in \tau$ and ${\chi }_1<{\chi }_2,$ then

$$\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)\le \frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(\varrho \right)d\varrho \le \frac{\mathsf{\Theta }\left({\chi }_1\right)+\mathsf{\Theta }\left({\chi }_2\right)}{2}.$$

(6)

The following well-known inequality is called the Ostrowski inequality:

Theorem 2

(Ostrowski inequality [16]). Let $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ a differentiable function in the interval τ and let ${\chi }_1,{\chi }_2\in \tau$ with ${\chi }_1<{\chi }_2.$ If $|{\mathsf{\Theta }}^{\prime }\left(x\right)|\le M$ for all $x\in [{\chi }_1,{\chi }_2],$ then

$$\left|\mathsf{\Theta }\left(x\right)-\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(\varrho \right)d\varrho \right|\le M({\chi }_2-{\chi }_1)\left[\frac{1}{4}+\frac{{\left(x-\frac{{\chi }_1+{\chi }_2}{2}\right)}^2}{{({\chi }_2-{\chi }_1)}^2}\right].$$

(7)

Another type of inequality is obtained by Dragomir et al. [25], which is as follows:

Theorem 3

(Simpson inequality [25]). Assume that $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ is a four-time continuous and differentiable function on $({\chi }_1,{\chi }_2)$ such that $\parallel {\mathsf{\Theta }}^{\left(4\right)}{\parallel }_{\infty }:={sup}_{x\in ({\chi }_1,{\chi }_2)}|{\mathsf{\Theta }}^{\left(4\right)}\left(x\right)|<\infty$ with ${\chi }_1<{\chi }_2,$ then

$$\left|\frac{1}{6}\left[\mathsf{\Theta }\left({\chi }_1\right)+4\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)+\mathsf{\Theta }\left({\chi }_2\right)\right]-\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)dx\right|\le \frac{1}{2880}{({\chi }_2-{\chi }_1)}^4{\parallel {\mathsf{\Theta }}^{\left(4\right)}\parallel }_{\infty }.$$

(8)

For brevity, we denote by $\mathcal{D}=\{h_1,h_2,\dots ,h_n,g_1,g_2,\dots ,g_m\}$ the convex set in the sequel.

Motivated by the above results, we introduce the following generic class of convex functions:

Definition 6.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$. A function $\mathsf{\Theta }:\tau \to \mathrm{R}$, which is nonnegative, is said to be $(n,m)$–generalized convex with respect to $\mathcal{D},$ if

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right),$$

(9)

holds for every ${\chi }_1,{\chi }_2\in \tau$ and $\varrho \in [0,1]$.

Remark 1.

From Definition 6, we can observe that:

1.

If $n=m=1,$$h_{{\ell }_1}\left(\varrho \right)=1-{(1-\varrho )}^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)=1-{\varrho }^{{\ell }_2},$ then we have Definition 1.

2.

If $n=m=1,$$h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s,$ then we obtain Definition 2.

3.

If $n=m=1,$$h_{{\ell }_1}\left(\varrho \right)=h\left(\varrho \right)$ and $g_{{\ell }_2}\left(\varrho \right)=h(1-\varrho ),$ then we obtain Definition 3.

4.

If $n=m,$ $h_{{\ell }_1}\left(\varrho \right)=1-{(1-\varrho )}^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)=1-{\varrho }^{{\ell }_2},$ then we obtain Definition 4.

5.

If $n=m,$ $h_{{\ell }_1}\left(\varrho \right)=1-{\left(s(1-\varrho )\right)}^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)=1-{\left(s\varrho \right)}^{{\ell }_2},$ then we obtain Definition 5.

Interested readers can derive many other known and unknown classes for suitable choices of the above functions $h_{{\ell }_1}$ and $g_{{\ell }_2}.$

This article is divided into five sections: in Section 2, algebraic properties of the $(n,m)$–generalized convex function are presented. In Section 3, a new version of the $\mathtt{HH}$ inequality is presented; by using this definition, we will also derive the products of two functions of this type. In Section 4, we obtain general results by using the well-known identities of midpoint-type inequalities of Ostrowski and Simpson for our new defined convex functions; we obtain special cases from these. Section 5 concludes the article.

2. Algebraic Properties of the New Convex Function

This section deals with algebraic properties of our new definition.

Theorem 4.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta },{\mathsf{\Theta }}_1,{\mathsf{\Theta }}_2:\tau \to \mathrm{R}.$ If $\mathsf{\Theta },{\mathsf{\Theta }}_1$, and ${\mathsf{\Theta }}_2$ are three nonnegative $(n,m)$–generalized convex functions with respect to $\mathcal{D},$ then

 1.

${\mathsf{\Theta }}_1+{\mathsf{\Theta }}_2$ is the $(n,m)$–generalized convex with respect to $\mathcal{D}$;

 2.

$c\mathsf{\Theta }$ is the $(n,m)$–generalized convex with respect to $\mathcal{D}$ for any nonnegative real number c.

Proof. 

The proof is evident, so we omit it. □

Theorem 5.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$. Let ${\mathsf{\Theta }}_1:\tau \to \mathrm{R}$ be a convex function and ${\mathsf{\Theta }}_2:\mathrm{R}\to \mathrm{R}$ is a non-decreasing and nonnegative $(n,m)$–generalized convex function with respect to $\mathcal{D}.$ Then the function ${\mathsf{\Theta }}_2\circ {\mathsf{\Theta }}_1:\tau \to \mathrm{R}$ is an $(n,m)$–generalized convex with respect to $\mathcal{D}.$

Proof. 

For all ${\chi }_1,{\chi }_2\in \tau$ and $\varrho \in [0,1],$ we have

$$\begin{array}{cc}\hfill \left({\mathsf{\Theta }}_2\circ {\mathsf{\Theta }}_1\right)(\varrho {\chi }_1+(1-\varrho ){\chi }_2)& ={\mathsf{\Theta }}_2\left({\mathsf{\Theta }}_1(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\right)\hfill \\ & \le {\mathsf{\Theta }}_2\left(\varrho {\mathsf{\Theta }}_1\left({\chi }_1\right)+\left(1-\varrho \right){\mathsf{\Theta }}_1\left({\chi }_2\right)\right)\hfill \\ & \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right){\mathsf{\Theta }}_2\left({\mathsf{\Theta }}_1\left({\chi }_1\right)\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){\mathsf{\Theta }}_2\left({\mathsf{\Theta }}_1\left({\chi }_2\right)\right)\hfill \\ & =\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)({\mathsf{\Theta }}_2\circ {\mathsf{\Theta }}_1)\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)({\mathsf{\Theta }}_2\circ {\mathsf{\Theta }}_1)\left({\chi }_2\right),\hfill \end{array}$$

which ends our proof. □

Theorem 6.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$. Let ${\mathsf{\Theta }}_k:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a family of nonnegative $(n,m)$–generalized convex functions with respect to $\mathcal{D}$ and $\mathsf{\Theta }\left(\chi \right)={sup}_k{\mathsf{\Theta }}_k\left(\chi \right)$. Then Θ is an $(n,m)$–generalized convex function with respect to $\mathcal{D}$ and $\mathcal{U}=\{\chi \in [{\chi }_1,{\chi }_2]:\mathsf{\Theta }\left(\chi \right)<+\infty \}$ is an interval.

Proof. 

Let ${\chi }_1,{\chi }_2\in \mathcal{U}$ and $\varrho \in [0,1],$ then

$$\begin{array}{cc}\hfill \mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)& =\underset{k}{sup}{\mathsf{\Theta }}_k(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\hfill \\ & \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\underset{k}{sup}{\mathsf{\Theta }}_k\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\underset{k}{sup}{\mathsf{\Theta }}_k\left({\chi }_2\right)\hfill \\ & =\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)<+\infty ,\hfill \end{array}$$

which ends our proof. □

Theorem 7.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$. If $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ is a nonnegative $(n,m)$–generalized convex function with respect to $\mathcal{D},$ then Θ is bounded on $[{\chi }_1,{\chi }_2].$

Proof. 

Let $K=max\{\mathsf{\Theta }\left({\chi }_1\right),\mathsf{\Theta }\left({\chi }_2\right)\}$ and $x\in [{\chi }_1,{\chi }_2]$. Then, there exists $\varrho \in [0,1]$, such that $x=\varrho {\chi }_1+(1-\varrho ){\chi }_2$. Moreover, since $h_{{\ell }_1},g_{{\ell }_2}$ are continuous functions on $[0,1]$ for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m,$ then we denote, respectively, $L_1=max\{h_1,h_2,\dots ,h_n\}$ and $L_2=max\{g_1,g_2,\dots ,g_m\}.$ Hence,

$$\begin{array}{cc}\hfill \mathsf{\Theta }\left(x\right)=\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)& \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)\hfill \\ & \le K\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)+\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right]\hfill \\ & \le K\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{L}_1+\frac{1}{m}\sum _{{\ell }_2=1}^m{L}_2\right]\hfill \\ & =K\left(L_1+L_2\right)=M.\hfill \end{array}$$

Moreover, for all $x\in [{\chi }_1,{\chi }_2],$ there exists $\xi \in \left[0,\frac{{\chi }_2-{\chi }_1}{2}\right]$, such that $x=\frac{{\chi }_1+{\chi }_2}{2}+\xi$ or $x=\frac{{\chi }_1+{\chi }_2}{2}-\xi .$ Let us suppose that $x=\frac{{\chi }_1+{\chi }_2}{2}+\xi$ without loss of generality. So, we have

$$\begin{array}{cc}\hfill \mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)& =\mathsf{\Theta }\left(\frac{1}{2}\left[\frac{{\chi }_1+{\chi }_2}{2}+\xi \right]+\frac{1}{2}\left[\frac{{\chi }_1+{\chi }_2}{2}-\xi \right]\right)\hfill \\ & \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)\right)\mathsf{\Theta }\left(x\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)\right)\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}-\xi \right)\hfill \\ & \le L_1\mathsf{\Theta }\left(x\right)+L_2\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}-\xi \right).\hfill \end{array}$$

By making use of M as the upper bound, we can deduce

$$\begin{array}{cc}\hfill & \mathsf{\Theta }\left(x\right)\ge \frac{1}{L_1}\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)-M=m,\hfill \end{array}$$

which ends our proof. □

3. The $\mathtt{HH}$ Inequality for the New Convex Function

In this section, we will establish some integral inequalities of the $\mathtt{HH}$-type pertaining to the $(n,m)$–generalized convex functions.

Theorem 8.

Assume that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$. If $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ is a nonnegative $(n,m)$–generalized convex function with respect to $\mathcal{D},$ then we have

$$\begin{array}{c}\frac{1}{\frac{1}{n}{\sum }_{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)+\frac{1}{m}{\sum }_{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)}\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)\le \frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)dx\hfill \\ \hfill \le \left(\frac{\mathsf{\Theta }\left({\chi }_1\right)+\mathsf{\Theta }\left({\chi }_2\right)}{2}\right)\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}+\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right],\end{array}$$

(10)

where

$$H_{{\ell }_1}:={\int }_0^1{h}_{{\ell }_1}\left(\varrho \right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n\mathrm{and}{G}_{{\ell }_2}:={\int }_0^1{g}_{{\ell }_2}\left(\varrho \right)d\varrho ,\forall {\ell }_2=1,2,\dots ,m.$$

Proof. 

Let $w_1,w_2\in [{\chi }_1,{\chi }_2].$ Applying the $(n,m)$–generalized convexity with respect to $\mathcal{D}$ of $\mathsf{\Theta }$ on $[{\chi }_1,{\chi }_2],$ we have

$$\mathsf{\Theta }\left(\frac{w_1+w_2}{2}\right)\le \frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)\mathsf{\Theta }\left(w_1\right)+\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)\mathsf{\Theta }\left(w_2\right).$$

(11)

Let us denote, respectively, $w_1=\varrho {\chi }_2+(1-\varrho ){\chi }_1$ and $w_2=\varrho {\chi }_1+(1-\varrho ){\chi }_2.$ From inequality (11), we obtain

$$\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)\le \frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)\mathsf{\Theta }(\varrho {\chi }_2+(1-\varrho ){\chi }_1)+\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2).$$

(12)

Integrating on both sides (12), with respect to $\varrho$ from 0 to $1,$ we obtain

$$\begin{array}{cc}\hfill \mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)& \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)\right){\int }_0^1\mathsf{\Theta }(\varrho {\chi }_2+(1-\varrho ){\chi }_1)d\varrho \hfill \\ \hspace{1em}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)\right){\int }_0^1\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)d\varrho \\ & =\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{1}{2}\right)+\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{1}{2}\right)\right)\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)dx,\hfill \end{array}$$

which gives the proof of the left hand side of (10). For the right hand side of (10), we use the definition of $(n,m)$–generalized convexity with respect to $\mathcal{D}$ of $\mathsf{\Theta },$ where $\varrho \in [0,1].$ Hence,

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right),$$

and

$$\mathsf{\Theta }(\varrho {\chi }_2+(1-\varrho ){\chi }_1)\le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right).$$

Adding both of them, we have

$$\begin{array}{cc}\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)+\mathsf{\Theta }(\varrho {\chi }_2+(1-\varrho ){\chi }_1)\hfill & \le \left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)\hfill \\ \hfill \hspace{1em}& +\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right).\hfill \end{array}$$

(13)

Integrating on both sides (13) with respect to $\varrho$ from 0 to $1,$ we obtain

$$\begin{array}{cc} & {\int }_0^1\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)d\varrho +{\int }_0^1\mathsf{\Theta }(\varrho {\chi }_2+(1-\varrho ){\chi }_1)d\varrho \hfill \\ & \hfill \le {\int }_0^1\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)\right]d\varrho \hfill \\ \hspace{1em}& \hfill \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_0^1\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)\right]d\varrho ,\end{array}$$

which leads to

$$\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)dx\le \left(\frac{\mathsf{\Theta }\left({\chi }_1\right)+\mathsf{\Theta }\left({\chi }_2\right)}{2}\right)\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}+\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right],$$

which ends our proof. □

Remark 2.

We have particular cases from Theorem 8:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=\varrho$ for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, we have Theorem 1.
• If $n=m$, $h_{{\ell }_1}\left(\varrho \right)=1-{\left(s(1-\varrho )\right)}^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)=1-{\left(s\varrho \right)}^{{\ell }_2}$ for $s\in [0,1]$, ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, we obtain ([23], Theorem 2.1).
• If $n=m$, $h_{{\ell }_1}\left(\varrho \right)=1-{(1-\varrho )}^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)=1-{\varrho }^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, we obtain ([22], Theorem 4).

Theorem 9.

Let $1\le n_1\le m_1$ and $1\le n_2\le m_2$ where $n_1,n_2,m_1,m_2\in \mathbb{N}.$ Assume that $h_{{\ell }_1}^{\left(1\right)},g_{{\ell }_2}^{\left(1\right)},h_k^{\left(2\right)},g_l^{\left(2\right)}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,\dots ,n_1,{\ell }_2=1,\dots ,m_1,k=1,\dots ,n_2$ and $l=1,\dots ,m_2.$ If $\mathsf{\Theta },\psi :[{\chi }_1,{\chi }_2]\to \mathrm{R}$ are nonnegative $(n_1,m_1)$ and $(n_2,m_2)$–generalized convex functions with respect to

${\mathcal{D}}^{\left(1\right)}=\left\{h_1^{\left(1\right)},h_2^{\left(1\right)},\dots ,h_{n_1}^{\left(1\right)},g_1^{\left(1\right)},g_2^{\left(1\right)},\dots ,g_{m_1}^{\left(1\right)}\right\}$, and

${\mathcal{D}}^{\left(2\right)}=\left\{h_1^{\left(2\right)},h_2^{\left(2\right)},\dots ,h_{n_2}^{\left(2\right)},g_1^{\left(2\right)},g_2^{\left(2\right)},\dots ,g_{m_2}^{\left(2\right)}\right\},$ respectively, then we have

$$\begin{array}{c}\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)\psi \left(x\right)dx\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\le \left(\frac{1}{n_1{n}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{k=1}^{n_2}{A}_{{\ell }_1,k}\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_1\right)+\left(\frac{1}{n_1{m}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{l=1}^{m_2}{B}_{{\ell }_1,l}\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{1}{n_2{m}_1}\sum _{k=1}^{n_2}\sum _{{\ell }_2=1}^{m_1}{C}_{k,{\ell }_2}\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_1\right)+\left(\frac{1}{m_1{m}_2}\sum _{{\ell }_2=1}^{m_1}\sum _{l=1}^{m_2}{D}_{{\ell }_2,l}\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_2\right),\hfill \end{array}$$

(14)

where

$$A_{{\ell }_1,k}:={\int }_0^1{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)h_k^{\left(2\right)}\left(\varrho \right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n_1,\forall k=1,2,\dots ,n_2,$$

$$B_{{\ell }_1,l}:={\int }_0^1{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)g_l^{\left(2\right)}\left(\varrho \right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n_1,\forall l=1,2,\dots ,m_2,$$

$$C_{k,{\ell }_2}:={\int }_0^1{h}_k^{\left(2\right)}\left(\varrho \right)g_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)d\varrho ,\forall k=1,2,\dots ,n_2,\forall {\ell }_2=1,2,\dots ,m_1,$$

and

$$D_{{\ell }_2,l}:={\int }_0^1{g}_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)g_l^{\left(2\right)}\left(\varrho \right)d\varrho ,\forall {\ell }_2=1,2,\dots ,m_1,\forall l=1,2,\dots ,m_2.$$

Proof. 

Applying $(n_1,m_1)$ and $(n_2,m_2)$–generalized convexity with respect to ${\mathcal{D}}^{\left(1\right)}$ and ${\mathcal{D}}^{\left(2\right)}$ of $\mathsf{\Theta },\psi$ on $[{\chi }_1,{\chi }_2],$ respectively, we have

$$\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \left(\frac{1}{n_1}\sum _{{\ell }_1=1}^{n_1}{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)+\left(\frac{1}{m_1}\sum _{{\ell }_2=1}^{m_1}{g}_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)$$

(15)

and

$$\psi (\varrho {\chi }_1+(1-\varrho ){\chi }_2)\le \left(\frac{1}{n_2}\sum _{k=1}^{n_2}{h}_k^{\left(2\right)}\left(\varrho \right)\right)\psi \left({\chi }_1\right)+\left(\frac{1}{m_2}\sum _{l=1}^{m_2}{g}_l^{\left(2\right)}\left(\varrho \right)\right)\psi \left({\chi }_2\right).$$

(16)

Multiplying inequalities (15) and (16) on both sides, we obtain

$$\begin{array}{c}\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\psi (\varrho {\chi }_1+(1-\varrho ){\chi }_2)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le \left(\frac{1}{n_1}\sum _{{\ell }_1=1}^{n_1}{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)\right)\left(\frac{1}{n_2}\sum _{k=1}^{n_2}{h}_k^{\left(2\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_1\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{1}{n_1}\sum _{{\ell }_1=1}^{n_1}{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)\right)\left(\frac{1}{m_2}\sum _{l=1}^{m_2}{g}_l^{\left(2\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_2\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{1}{n_2}\sum _{k=1}^{n_2}{h}_k^{\left(2\right)}\left(\varrho \right)\right)\left(\frac{1}{m_1}\sum _{{\ell }_2=1}^{m_1}{g}_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_1\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\left(\frac{1}{m_1}\sum _{{\ell }_2=1}^{m_1}{g}_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)\right)\left(\frac{1}{m_2}\sum _{l=1}^{m_2}{g}_l^{\left(2\right)}\left(\varrho \right)\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_2\right).\hfill \end{array}$$

(17)

Integrating inequality (17) with respect to $\varrho$ from 0 to 1 on both sides, we obtain

$${\int }_0^1\mathsf{\Theta }(\varrho {\chi }_1+(1-\varrho ){\chi }_2)\psi (\varrho {\chi }_1+(1-\varrho ){\chi }_2)d\varrho =\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(x\right)\psi \left(x\right)dx$$

$$\le \left(\frac{1}{n_1{n}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{k=1}^{n_2}{\int }_0^1{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)h_k^{\left(2\right)}\left(\varrho \right)d\varrho \right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_1\right)$$

$$+\left(\frac{1}{n_1{m}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{l=1}^{m_2}{\int }_0^1{h}_{{\ell }_1}^{\left(1\right)}\left(\varrho \right)g_l^{\left(2\right)}\left(\varrho \right)d\varrho \right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_2\right)$$

$$+\left(\frac{1}{n_2{m}_1}\sum _{k=1}^{n_2}\sum _{{\ell }_2=1}^{m_1}{\int }_0^1{h}_k^{\left(2\right)}\left(\varrho \right)g_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)d\varrho \right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_1\right)$$

$$+\left(\frac{1}{m_1{m}_2}\sum _{{\ell }_2=1}^{m_1}\sum _{l=1}^{m_2}{\int }_0^1{g}_{{\ell }_2}^{\left(1\right)}\left(\varrho \right)g_l^{\left(2\right)}\left(\varrho \right)d\varrho \right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_2\right).$$

$$=\left(\frac{1}{n_1{n}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{k=1}^{n_2}{A}_{{\ell }_1,k}\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_1\right)+\left(\frac{1}{n_1{m}_2}\sum _{{\ell }_1=1}^{n_1}\sum _{l=1}^{m_2}{B}_{{\ell }_1,l}\right)\mathsf{\Theta }\left({\chi }_1\right)\psi \left({\chi }_2\right)$$

$$+\left(\frac{1}{n_2{m}_1}\sum _{k=1}^{n_2}\sum _{{\ell }_2=1}^{m_1}{C}_{k,{\ell }_2}\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_1\right)+\left(\frac{1}{m_1{m}_2}\sum _{{\ell }_2=1}^{m_1}\sum _{l=1}^{m_2}{D}_{{\ell }_2,l}\right)\mathsf{\Theta }\left({\chi }_2\right)\psi \left({\chi }_2\right),$$

which ends our proof. □

4. Further Results

We denote by $\mathcal{L}[{\chi }_1,{\chi }_2]$ the set of all integrable functions on $[{\chi }_1,{\chi }_2].$ Let us recall the following lemmas in order to establish our following results.

Lemma 1

(Midpoint identity [26]). Let $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ be a differentiable function on τ and ${\chi }_1,{\chi }_2\in \tau$ with ${\chi }_1<{\chi }_2.$ If ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2],$ then

$$\begin{array}{c}\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2):=\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(\varrho \right)d\varrho -\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{({\chi }_2-{\chi }_1)}{4}\left\{{\int }_0^1\varrho {\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{2-\varrho }{2}{\chi }_2\right)d\varrho -{\int }_0^1\varrho {\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{2-\varrho }{2}{\chi }_1\right)d\varrho \right\}.\hfill \end{array}$$

(18)

Lemma 2

(Ostrowski identity [27]). Let $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ be a differentiable function on τ and ${\chi }_1,{\chi }_2\in \tau$ with ${\chi }_1<{\chi }_2.$ If ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2],$ then

$$\begin{array}{c}{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2):=\mathsf{\Theta }\left(x\right)-\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(\varrho \right)d\varrho \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho {\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)d\varrho -\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho {\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)d\varrho .\hfill \end{array}$$

(19)

Lemma 3

(Simpson identity [28]). Let $\mathsf{\Theta }:\tau \subseteq \mathrm{R}\to \mathrm{R}$ be a differentiable function on τ and ${\chi }_1,{\chi }_2\in \tau$ with ${\chi }_1<{\chi }_2.$ If ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2],$ then

$$\begin{array}{c}{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2):=\frac{1}{6}\left[\mathsf{\Theta }\left({\chi }_1\right)+4\mathsf{\Theta }\left(\frac{{\chi }_1+{\chi }_2}{2}\right)+\mathsf{\Theta }\left({\chi }_2\right)\right]-\frac{1}{{\chi }_2-{\chi }_1}{\int }_{{\chi }_1}^{{\chi }_2}\mathsf{\Theta }\left(\varrho \right)d\varrho \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=({\chi }_2-{\chi }_1)\left\{{\int }_0^{\frac{1}{2}}\left(\varrho -\frac{1}{6}\right){\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)d\varrho +{\int }_{\frac{1}{2}}^1\left(\varrho -\frac{5}{6}\right){\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)d\varrho \right\}.\hfill \end{array}$$

(20)

Theorem 10.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$ such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }|$ is an $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then we have

$$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left(|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right)\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}+\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right],$$

(21)

where

$$U_{{\ell }_1}:={\int }_0^1\varrho h_{{\ell }_1}\left(\frac{\varrho }{2}\right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n\mathrm{and}{V}_{{\ell }_2}:={\int }_0^1\varrho g_{{\ell }_2}\left(\frac{\varrho }{2}\right)d\varrho ,\forall {\ell }_2=1,2,\dots ,m.$$

Proof. 

By using Lemma 1 and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }|$ with respect to $\mathcal{D}$, we have

$$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left\{{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{(2-\varrho )}{2}{\chi }_2\right)\right|d\varrho +{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{(2-\varrho )}{2}{\chi }_1\right)\right|d\varrho \right\}$$

$$\le \frac{({\chi }_2-{\chi }_1)}{4}\{{\int }_0^1\varrho \left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right]d\varrho$$

$$+{\int }_0^1\varrho \left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]d\varrho \}$$

$$=\frac{({\chi }_2-{\chi }_1)}{4}\left(|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right)\left[\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}+\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right],$$

which ends our proof. □

Corollary 1.

We have particular cases from Theorem 10:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{8}\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right].$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right]\hfill \\ \hfill \times \left[\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{2^{{\ell }_1}({\ell }_1+2)}+\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{2^{{\ell }_2}}\left(\frac{2}{{\ell }_2+1}\left(2^{{\ell }_2+1}-1\right)-\frac{1}{{\ell }_2+2}\left(2^{{\ell }_2+2}-1\right)\right)\right].\end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right]\hfill \\ \hfill \times \left[\frac{1}{2^s(s+2)}+\frac{1}{2^s}\left(\frac{2}{s+1}\left(2^{s+1}-1\right)-\frac{1}{s+2}\left(2^{s+2}-1\right)\right)\right].\end{array}$$

Theorem 11.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is an $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2]$, then for $q>1$ and $\frac{1}{p}+\frac{1}{q}=1$, we have

$$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{M}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{N}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill +{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{M}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{N}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

(22)

where

$$M_{{\ell }_1}:={\int }_0^1{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n\mathrm{and}{N}_{{\ell }_2}:={\int }_0^1{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)d\varrho ,\forall {\ell }_2=1,2,\dots ,m.$$

Proof. 

By using Lemma 1, Hölder’s inequality and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we have

$$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left\{{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{(2-\varrho )}{2}{\chi }_2\right)\right|d\varrho +{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{(2-\varrho )}{2}{\chi }_1\right)\right|d\varrho \right\}$$

$$\le \frac{({\chi }_2-{\chi }_1)}{4}{\left({\int }_0^1{\varrho }^{p}d\varrho \right)}^{\frac{1}{p}}$$

$$\times \left\{{\left({\int }_0^1{\left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{(2-\varrho )}{2}{\chi }_2\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}+{\left({\int }_0^1{\left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{(2-\varrho )}{2}{\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}\right\}$$

$$\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}$$

$$\times \{{\left[{\int }_0^1\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$+{\left[{\int }_0^1\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}\}$$

$$=\frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{M}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{N}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{M}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{N}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},$$

which ends our proof. □

Corollary 2.

We have particular cases from Theorem 11:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we have

  $$\begin{array}{c}\hfill \left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4\sqrt[q]{4}}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left\{{\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+3{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}+{\left[3|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.\end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{2^{{\ell }_1}({\ell }_1+1)}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{2^{{\ell }_2+1}-1}{2^{{\ell }_2}({\ell }_2+1)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{2^{{\ell }_1}({\ell }_1+1)}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{2^{{\ell }_2+1}-1}{2^{{\ell }_2}({\ell }_2+1)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\}.\hfill \end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4\sqrt[q]{2^s(s+1)}}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\times \left\{{\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+(2^{s+1}-1){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}+{\left[(2^{s+1}-1)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.\hfill \end{array}$$

Theorem 12.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is an $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2]$, then for $q>1,$ we have

$$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill +{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

(23)

where $U_{{\ell }_1}$ and $V_{{\ell }_2}$ are defined as in Theorem 10.

Proof. 

By using Lemma 1, the well-known power mean inequality and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we have

$$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}\left\{{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{(2-\varrho )}{2}{\chi }_2\right)\right|d\varrho +{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{(2-\varrho )}{2}{\chi }_1\right)\right|d\varrho \right\}$$

$$\le \frac{({\chi }_2-{\chi }_1)}{4}{\left({\int }_0^1\varrho d\varrho \right)}^{1-\frac{1}{q}}$$

$$\times \left\{{\left({\int }_0^1\varrho {\left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_1+\frac{(2-\varrho )}{2}{\chi }_2\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}+{\left({\int }_0^1\varrho {\left|{\mathsf{\Theta }}^{\prime }\left(\frac{\varrho }{2}{\chi }_2+\frac{(2-\varrho )}{2}{\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}\right\}$$

$$\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}$$

$$\times \{{\left[{\int }_0^1\varrho \left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$+{\left[{\int }_0^1\varrho \left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\frac{\varrho }{2}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\frac{\varrho }{2}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}\}$$

$$=\frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{U}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{V}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},$$

which ends our proof. □

Corollary 3.

We have particular cases from Theorem 12:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{8\sqrt[q]{3}}\left\{{\left[|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+2{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}+{\left[2|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\hfill \\ \times \left\{\right[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{2^{{\ell }_1}({\ell }_1+2)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\\ +\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{2^{{\ell }_2}}\left(\frac{2}{{\ell }_2+1}\left(2^{{\ell }_2+1}-1\right)-\frac{1}{{\ell }_2+2}\left(2^{{\ell }_2+2}-1\right)\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q{]}^{\frac{1}{q}}\\ +[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{2^{{\ell }_1}({\ell }_1+2)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\\ \hfill +\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{2^{{\ell }_2}}\left(\frac{2}{{\ell }_2+1}\left(2^{{\ell }_2+1}-1\right)-\frac{1}{{\ell }_2+2}\left(2^{{\ell }_2+2}-1\right)\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q{]}^{\frac{1}{q}}\}.\end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|\mathcal{T}(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le \frac{({\chi }_2-{\chi }_1)}{4}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\hfill \\ \times \{{\left[\frac{1}{2^s(s+2)}|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+\frac{1}{2^s}\left(\frac{2}{s+1}\left(2^{s+1}-1\right)-\frac{1}{s+2}\left(2^{s+2}-1\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\left[\frac{1}{2^s(s+2)}|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\frac{1}{2^s}\left(\frac{2}{s+1}\left(2^{s+1}-1\right)-\frac{1}{s+2}\left(2^{s+2}-1\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\}.\end{array}$$

Theorem 13.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }|$ is an $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then we have

$$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]\hfill \\ \hfill +\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right],\end{array}$$

(24)

where

$$E_{{\ell }_1}:={\int }_0^1\varrho h_{{\ell }_1}\left(\varrho \right)d\varrho ,\forall {\ell }_1=1,2,\dots ,n\mathrm{and}{F}_{{\ell }_2}:={\int }_0^1\varrho g_{{\ell }_2}\left(\varrho \right)d\varrho ,\forall {\ell }_2=1,2,\dots ,m.$$

Proof. 

By using Lemma 2 and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }|$ with respect to $\mathcal{D}$, we have

$$\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)\right|d\varrho$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)\right|d\varrho$$

$$\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]d\varrho$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right]d\varrho$$

$$=\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right],$$

which ends our proof. □

Corollary 4.

We have particular cases from Theorem 13:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{6({\chi }_2-{\chi }_1)}\left[2|{\mathsf{\Theta }}^{\prime }\left(x\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]+\frac{{({\chi }_2-x)}^2}{6({\chi }_2-{\chi }_1)}\left[2|{\mathsf{\Theta }}^{\prime }\left(x\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right].$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}‘\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+2}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{({\ell }_2+1)({\ell }_2+2)}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]\hfill \\ \hfill +\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+2}\right)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{({\ell }_2+1)({\ell }_2+2)}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right].\end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{1}{(s+1)(s+2)}\hfill \\ \hfill \times \left\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}\left[(s+1)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}\left[(s+1)|{\mathsf{\Theta }}^{\prime }\left(x\right)|+|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\right]\right\}.\end{array}$$

Theorem 14.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$ such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then for $q>1$ and $\frac{1}{p}+\frac{1}{q}=1,$ we obtain

$$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\},\hfill \end{array}$$

(25)

where $H_{{\ell }_1}$ and $G_{{\ell }_2}$ are defined as in Theorem 8.

Proof. 

By using Lemma 2, Hölder’s inequality and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we obtain

$$\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)\right|d\varrho$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)\right|d\varrho$$

$$\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left({\int }_0^1{\varrho }^{p}d\varrho \right)}^{\frac{1}{p}}{\left({\int }_0^1{\left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left({\int }_0^1{\varrho }^{p}d\varrho \right)}^{\frac{1}{p}}{\left({\int }_0^1{\left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}$$

$$\le {\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[{\int }_0^1\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[{\int }_0^1\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}\}$$

$$={\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\},$$

which ends our proof. □

Corollary 5.

We have particular cases from Theorem 14:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{2}\right)}^{\frac{1}{q}}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.\hfill \end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{{\ell }_2+1}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{{\ell }_2+1}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\}.\hfill \end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{s+1}\right)}^{\frac{1}{q}}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \left\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.\hfill \end{array}$$

Theorem 15.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is the $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then for $q>1,$ we have

$$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\hfill \\ \hfill +\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

(26)

where $E_{{\ell }_1}$ and $F_{{\ell }_2}$ are defined as in Theorem 13.

Proof. 

By using Lemma 2, the well-known power mean inequality and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we have

$$\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)\right|d\varrho$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\int }_0^1\varrho \left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)\right|d\varrho$$

$$\le \frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left({\int }_0^1\varrho d\varrho \right)}^{1-\frac{1}{q}}{\left({\int }_0^1\varrho {\left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left({\int }_0^1\varrho d\varrho \right)}^{1-\frac{1}{q}}{\left({\int }_0^1\varrho {\left|{\mathsf{\Theta }}^{\prime }\left(\varrho x+(1-\varrho ){\chi }_2\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}$$

$$\le {\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[{\int }_0^1\varrho \left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[{\int }_0^1\varrho \left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}\}$$

$$={\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{E}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{F}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\},$$

which ends our proof. □

Corollary 6.

We have particular cases from Theorem 15:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le \frac{1}{2\sqrt[q]{3}}$$

  $$\times \left\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[2|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[2|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\hfill \\ \times \{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+2}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{({\ell }_2+1)({\ell }_2+2)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\\ \hfill +\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+2}\right)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{({\ell }_2+1)({\ell }_2+2)}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\}.\end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_1(\mathsf{\Theta };x,{\chi }_1,{\chi }_2)\right|\le {\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}{\left(\frac{1}{(s+1)(s+2)}\right)}^{\frac{1}{q}}\hfill \\ \hfill \times \left\{\frac{{(x-{\chi }_1)}^2}{{\chi }_2-{\chi }_1}{\left[(s+1)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}+\frac{{({\chi }_2-x)}^2}{{\chi }_2-{\chi }_1}{\left[(s+1)|{\mathsf{\Theta }}^{\prime }{\left(x\right)|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q\right]}^{\frac{1}{q}}\right\}.\end{array}$$

Theorem 16.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }|$ is the $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then we have

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\times \{\frac{1}{n}\sum _{{\ell }_1=1}^n\left[\left(A_{{\ell }_1}^{\left(1\right)}+A_{{\ell }_1}^{\left(3\right)}\right)-\left(A_{{\ell }_1}^{\left(2\right)}+A_{{\ell }_1}^{\left(4\right)}\right)+\left(B_{{\ell }_1}^{\left(2\right)}+B_{{\ell }_1}^{\left(4\right)}\right)-\left(B_{{\ell }_1}^{\left(1\right)}+B_{{\ell }_1}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{m}\sum _{{\ell }_2=1}^m\left[\left(C_{{\ell }_2}^{\left(1\right)}+C_{{\ell }_2}^{\left(3\right)}\right)-\left(C_{{\ell }_2}^{\left(2\right)}+C_{{\ell }_2}^{\left(4\right)}\right)+\left(D_{{\ell }_2}^{\left(2\right)}+D_{{\ell }_2}^{\left(4\right)}\right)-\left(D_{{\ell }_2}^{\left(1\right)}+D_{{\ell }_2}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\},\hfill \end{array}$$

(27)

where

$$A_{{\ell }_1}^{\left(1\right)}:={\int }_0^{\frac{1}{6}}{h}_{{\ell }_1}\left(\varrho \right)d\varrho ,A_{{\ell }_1}^{\left(2\right)}:={\int }_{\frac{1}{6}}^{\frac{1}{2}}{h}_{{\ell }_1}\left(\varrho \right)d\varrho ,$$

$$A_{{\ell }_1}^{\left(3\right)}:={\int }_{\frac{1}{2}}^{\frac{5}{6}}{h}_{{\ell }_1}\left(\varrho \right)d\varrho ,A_{{\ell }_1}^{\left(4\right)}:={\int }_{\frac{5}{6}}^1{h}_{{\ell }_1}\left(\varrho \right)d\varrho ,$$

$$B_{{\ell }_1}^{\left(1\right)}:={\int }_0^{\frac{1}{6}}\varrho h_{{\ell }_1}\left(\varrho \right)d\varrho ,B_{{\ell }_1}^{\left(2\right)}:={\int }_{\frac{1}{6}}^{\frac{1}{2}}\varrho h_{{\ell }_1}\left(\varrho \right)d\varrho ,$$

$$B_{{\ell }_1}^{\left(3\right)}:={\int }_{\frac{1}{2}}^{\frac{5}{6}}\varrho h_{{\ell }_1}\left(\varrho \right)d\varrho ,B_{{\ell }_1}^{\left(4\right)}:={\int }_{\frac{5}{6}}^1\varrho h_{{\ell }_1}\left(\varrho \right)d\varrho ,$$

and

$$C_{{\ell }_2}^{\left(1\right)}:={\int }_0^{\frac{1}{6}}{g}_{{\ell }_2}\left(\varrho \right)d\varrho ,C_{{\ell }_2}^{\left(2\right)}:={\int }_{\frac{1}{6}}^{\frac{1}{2}}{g}_{{\ell }_2}\left(\varrho \right)d\varrho ,$$

$$C_{{\ell }_2}^{\left(3\right)}:={\int }_{\frac{1}{2}}^{\frac{5}{6}}{g}_{{\ell }_2}\left(\varrho \right)d\varrho ,C_{{\ell }_2}^{\left(4\right)}:={\int }_{\frac{5}{6}}^1{g}_{{\ell }_2}\left(\varrho \right)d\varrho ,$$

$$D_{{\ell }_2}^{\left(1\right)}:={\int }_0^{\frac{1}{6}}\varrho g_{{\ell }_2}\left(\varrho \right)d\varrho ,D_{{\ell }_2}^{\left(2\right)}:={\int }_{\frac{1}{6}}^{\frac{1}{2}}\varrho g_{{\ell }_2}\left(\varrho \right)d\varrho ,$$

$$D_{{\ell }_2}^{\left(3\right)}:={\int }_{\frac{1}{2}}^{\frac{5}{6}}\varrho g_{{\ell }_2}\left(\varrho \right)d\varrho ,D_{{\ell }_2}^{\left(4\right)}:={\int }_{\frac{5}{6}}^1\varrho g_{{\ell }_2}\left(\varrho \right)d\varrho ,$$

for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$.

Proof. 

By using Lemma 3 and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }|$ with respect to $\mathcal{D}$, we have

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1)\{{\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \}\hfill \end{array}$$

$$\le ({\chi }_2-{\chi }_1)\{{\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]d\varrho$$

$$+{\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\right]d\varrho \}$$

$$=({\chi }_2-{\chi }_1)\{\frac{1}{n}\sum _{{\ell }_1=1}^n\left[\left(A_{{\ell }_1}^{\left(1\right)}+A_{{\ell }_1}^{\left(3\right)}\right)-\left(A_{{\ell }_1}^{\left(2\right)}+A_{{\ell }_1}^{\left(4\right)}\right)+\left(B_{{\ell }_1}^{\left(2\right)}+B_{{\ell }_1}^{\left(4\right)}\right)-\left(B_{{\ell }_1}^{\left(1\right)}+B_{{\ell }_1}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|$$

$$+\frac{1}{m}\sum _{{\ell }_2=1}^m\left[\left(C_{{\ell }_2}^{\left(1\right)}+C_{{\ell }_2}^{\left(3\right)}\right)-\left(C_{{\ell }_2}^{\left(2\right)}+C_{{\ell }_2}^{\left(4\right)}\right)+\left(D_{{\ell }_2}^{\left(2\right)}+D_{{\ell }_2}^{\left(4\right)}\right)-\left(D_{{\ell }_2}^{\left(1\right)}+D_{{\ell }_2}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\},$$

which ends our proof. □

Corollary 7.

If we take $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1},\forall {\ell }_1=1,\dots ,n$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2},\forall {\ell }_2=1,\dots ,m$ in Theorem 16, we obtain

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1)\hfill \\ \times \{\frac{1}{n}\sum _{{\ell }_1=1}^n\left[\left(U_{{\ell }_1}^{\left(1\right)}+U_{{\ell }_1}^{\left(3\right)}\right)-\left(U_{{\ell }_1}^{\left(2\right)}+U_{{\ell }_1}^{\left(4\right)}\right)+\left(V_{{\ell }_1}^{\left(2\right)}+V_{{\ell }_1}^{\left(4\right)}\right)-\left(V_{{\ell }_1}^{\left(1\right)}+V_{{\ell }_1}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|\\ \hfill +\frac{1}{m}\sum _{{\ell }_2=1}^m\left[\left(P_{{\ell }_2}^{\left(1\right)}+P_{{\ell }_2}^{\left(3\right)}\right)-\left(P_{{\ell }_2}^{\left(2\right)}+P_{{\ell }_2}^{\left(4\right)}\right)+\left(Q_{{\ell }_2}^{\left(2\right)}+Q_{{\ell }_2}^{\left(4\right)}\right)-\left(Q_{{\ell }_2}^{\left(1\right)}+Q_{{\ell }_2}^{\left(3\right)}\right)\right]|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|\},\end{array}$$

where

$$U_{{\ell }_1}^{\left(1\right)}:=\frac{1}{6^{{\ell }_1+1}({\ell }_1+1)},U_{{\ell }_1}^{\left(2\right)}:=\frac{1}{{\ell }_1+1}\left(\frac{1}{2^{{\ell }_1+1}}-\frac{1}{6^{{\ell }_1+1}}\right),$$

$$U_{{\ell }_1}^{\left(3\right)}:=\frac{1}{{\ell }_1+1}\left({\left(\frac{5}{6}\right)}^{{\ell }_1+1}-\frac{1}{2^{{\ell }_1+1}}\right),U_{{\ell }_1}^{\left(4\right)}:=\frac{1}{{\ell }_1+1}\left(1-{\left(\frac{5}{6}\right)}^{{\ell }_1+1}\right),$$

$$V_{{\ell }_1}^{\left(1\right)}:=\frac{1}{6^{{\ell }_1+2}({\ell }_1+2)},V_{{\ell }_1}^{\left(2\right)}:=\frac{1}{{\ell }_1+2}\left(\frac{1}{2^{{\ell }_1+2}}-\frac{1}{6^{{\ell }_1+2}}\right),$$

$$V_{{\ell }_1}^{\left(3\right)}:=\frac{1}{{\ell }_1+2}\left({\left(\frac{5}{6}\right)}^{{\ell }_1+2}-\frac{1}{2^{{\ell }_1+2}}\right),V_{{\ell }_1}^{\left(4\right)}:=\frac{1}{{\ell }_1+2}\left(1-{\left(\frac{5}{6}\right)}^{{\ell }_1+2}\right),$$

$$P_{{\ell }_2}^{\left(1\right)}:=\frac{1}{{\ell }_2+1}\left(1-{\left(\frac{5}{6}\right)}^{{\ell }_2+1}\right),P_{{\ell }_2}^{\left(2\right)}:=\frac{1}{{\ell }_2+1}\left({\left(\frac{5}{6}\right)}^{{\ell }_2+1}-\frac{1}{2^{{\ell }_2+1}}\right),$$

$$P_{{\ell }_2}^{\left(3\right)}:=\frac{1}{{\ell }_2+1}\left(\frac{1}{2^{{\ell }_2+1}}-\frac{1}{6^{{\ell }_2+1}}\right),P_{{\ell }_2}^{\left(4\right)}:=\frac{1}{6^{{\ell }_2+1}({\ell }_2+1)},$$

and

$$Q_{{\ell }_2}^{\left(1\right)}:=\frac{1}{({\ell }_2+1)({\ell }_2+2)}-\left(\frac{1}{{\ell }_2+1}{\left(\frac{5}{6}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{5}{6}\right)}^{{\ell }_2+2}\right),$$

$$Q_{{\ell }_2}^{\left(2\right)}:=\left(\frac{1}{{\ell }_2+1}{\left(\frac{5}{6}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{5}{6}\right)}^{{\ell }_2+2}\right)-\left(\frac{1}{{\ell }_2+1}{\left(\frac{1}{2}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{1}{2}\right)}^{{\ell }_2+2}\right),$$

$$Q_{{\ell }_2}^{\left(3\right)}:=\left(\frac{1}{{\ell }_2+1}{\left(\frac{1}{2}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{1}{2}\right)}^{{\ell }_2+2}\right)-\left(\frac{1}{{\ell }_2+1}{\left(\frac{1}{6}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{1}{6}\right)}^{{\ell }_2+2}\right),$$

$$Q_{{\ell }_2}^{\left(4\right)}:=\frac{1}{{\ell }_2+1}{\left(\frac{1}{6}\right)}^{{\ell }_2+1}-\frac{1}{{\ell }_2+2}{\left(\frac{1}{6}\right)}^{{\ell }_2+2}.$$

Theorem 17.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$ such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is the $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then for $q>1$ and $\frac{1}{p}+\frac{1}{q}=1,$ we have

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left[\frac{2}{p+1}\left(\frac{2^{p+1}+1}{6^{p+1}}\right)\right]}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}},\hfill \end{array}$$

(28)

where $H_{{\ell }_1}$ and $G_{{\ell }_2}$ are as defined in Theorem 8.

Proof. 

By using Lemma 3, Hölder’s inequality and the $(n,m)$–generalized convexity of $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we have

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1)\{{\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \}\hfill \end{array}$$

$$\le ({\chi }_2-{\chi }_1)\left[{\left({\int }_0^{\frac{1}{2}}{\left|\varrho -\frac{1}{6}\right|}^{p}d\varrho \right)}^{\frac{1}{p}}+{\left({\int }_{\frac{1}{2}}^1{\left|\varrho -\frac{5}{6}\right|}^{p}d\varrho \right)}^{\frac{1}{p}}\right]{\left({\int }_0^1{\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}$$

$$\le ({\chi }_2-{\chi }_1)\left[{\left({\int }_0^{\frac{1}{2}}{\left|\varrho -\frac{1}{6}\right|}^{p}d\varrho \right)}^{\frac{1}{p}}+{\left({\int }_{\frac{1}{2}}^1{\left|\varrho -\frac{5}{6}\right|}^{p}d\varrho \right)}^{\frac{1}{p}}\right]$$

$$\times {\left[{\int }_0^1\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$=({\chi }_2-{\chi }_1){\left[\frac{2}{p+1}\left(\frac{2^{p+1}+1}{6^{p+1}}\right)\right]}^{\frac{1}{p}}{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{H}_{{\ell }_1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{G}_{{\ell }_2}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}},$$

which ends our proof. □

Corollary 8.

We have particular cases from Theorem 17:

• If $h_{{\ell }_1}\left(\varrho \right)=\varrho$ and $g_{{\ell }_2}\left(\varrho \right)=1-\varrho$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left[\frac{2}{p+1}\left(\frac{2^{p+1}+1}{6^{p+1}}\right)\right]}^{\frac{1}{p}}{\left[\frac{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q}{2}\right]}^{\frac{1}{q}}.$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ for all ${\ell }_1=1,2,\dots ,n,$ and ${\ell }_2=1,2,\dots ,m$, we obtain

  $$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left[\frac{2}{p+1}\left(\frac{2^{p+1}+1}{6^{p+1}}\right)\right]}^{\frac{1}{p}}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times {\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\frac{1}{{\ell }_1+1}\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\frac{1}{{\ell }_2+1}\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}.\hfill \end{array}$$
• If $h_{{\ell }_1}\left(\varrho \right)={\varrho }^s$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^s$ for all ${\ell }_1=1,\dots ,n$, ${\ell }_2=1,\dots ,m,$ and $s\in (0,1]$, we obtain

  $$\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left[\frac{2}{p+1}\left(\frac{2^{p+1}+1}{6^{p+1}}\right)\right]}^{\frac{1}{p}}{\left[\frac{|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right){|}^q+{|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right)|}^q}{s+1}\right]}^{\frac{1}{q}}.$$

Theorem 18.

Suppose that $1\le n\le m$, where $n,m\in \mathbb{N}$, and assume that $h_{{\ell }_1},g_{{\ell }_2}:[0,1]\to [0,+\infty )$ are continuous functions for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, and $\mathsf{\Theta }:[{\chi }_1,{\chi }_2]\to \mathrm{R}$ be a differentiable function on $({\chi }_1,{\chi }_2)$, such that ${\mathsf{\Theta }}^{\prime }\in \mathcal{L}[{\chi }_1,{\chi }_2].$ If $|{\mathsf{\Theta }}^{\prime }{|}^q$ is the $(n,m)$–generalized convex function with respect to $\mathcal{D}$ on $[{\chi }_1,{\chi }_2],$ then for $q>1,$ we have

$$\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left(\frac{5}{36}\right)}^{1-\frac{1}{q}}$$

(29)

$$\times \{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{A_{{\ell }_1}^{\left(1\right)}}{6}-B_{{\ell }_1}^{\left(1\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{C_{{\ell }_2}^{\left(1\right)}}{6}-D_{{\ell }_2}^{\left(1\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(B_{{\ell }_1}^{\left(2\right)}-\frac{A_{{\ell }_1}^{\left(2\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(D_{{\ell }_2}^{\left(2\right)}-\frac{C_{{\ell }_2}^{\left(2\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{5A_{{\ell }_1}^{\left(3\right)}}{6}-B_{{\ell }_1}^{\left(3\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{5C_{{\ell }_2}^{\left(3\right)}}{6}-D_{{\ell }_2}^{\left(3\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(B_{{\ell }_1}^{\left(4\right)}-\frac{5A_{{\ell }_1}^{\left(4\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(D_{{\ell }_2}^{\left(4\right)}-\frac{5C_{{\ell }_2}^{\left(4\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},$$

where $A_{{\ell }_1}^{\left(k\right)},B_{{\ell }_1}^{\left(k\right)},C_{{\ell }_2}^{\left(k\right)}$, and $D_{{\ell }_2}^{\left(k\right)}$ for all $k=1,2,3,4$ are defined as in Theorem 16.

Proof. 

By using Lemma 3, the well-known power mean inequality and $(n,m)$–generalized convexity $|{\mathsf{\Theta }}^{\prime }{|}^q$ with respect to $\mathcal{D}$, we have

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1)\{{\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \hfill \\ \hfill +{\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|d\varrho \}\end{array}$$

$$\le ({\chi }_2-{\chi }_1)\left[{\left({\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|d\varrho \right)}^{1-\frac{1}{q}}+{\left({\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|d\varrho \right)}^{1-\frac{1}{q}}\right]$$

$$\times \left\{{\left({\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|{\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}+{\left({\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|{\left|{\mathsf{\Theta }}^{\prime }\left(\varrho {\chi }_2+(1-\varrho ){\chi }_1\right)\right|}^{q}d\varrho \right)}^{\frac{1}{q}}\right\}$$

$$\le ({\chi }_2-{\chi }_1)\left[{\left({\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|d\varrho \right)}^{1-\frac{1}{q}}+{\left({\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|d\varrho \right)}^{1-\frac{1}{q}}\right]$$

$$\times \{{\left[{\int }_0^{\frac{1}{2}}\left|\varrho -\frac{1}{6}\right|\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}$$

$$+{\left[{\int }_{\frac{1}{2}}^1\left|\varrho -\frac{5}{6}\right|\left(\left(\frac{1}{n}\sum _{{\ell }_1=1}^n{h}_{{\ell }_1}\left(\varrho \right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m{g}_{{\ell }_2}\left(\varrho \right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right)d\varrho \right]}^{\frac{1}{q}}\}$$

$$=({\chi }_2-{\chi }_1){\left(\frac{5}{36}\right)}^{1-\frac{1}{q}}$$

$$\times \{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{A_{{\ell }_1}^{\left(1\right)}}{6}-B_{{\ell }_1}^{\left(1\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{C_{{\ell }_2}^{\left(1\right)}}{6}-D_{{\ell }_2}^{\left(1\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(B_{{\ell }_1}^{\left(2\right)}-\frac{A_{{\ell }_1}^{\left(2\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(D_{{\ell }_2}^{\left(2\right)}-\frac{C_{{\ell }_2}^{\left(2\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{5A_{{\ell }_1}^{\left(3\right)}}{6}-B_{{\ell }_1}^{\left(3\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{5C_{{\ell }_2}^{\left(3\right)}}{6}-D_{{\ell }_2}^{\left(3\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}$$

$$+{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(B_{{\ell }_1}^{\left(4\right)}-\frac{5A_{{\ell }_1}^{\left(4\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(D_{{\ell }_2}^{\left(4\right)}-\frac{5C_{{\ell }_2}^{\left(4\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},$$

which ends our proof. □

Corollary 9.

If we take $h_{{\ell }_1}\left(\varrho \right)={\varrho }^{{\ell }_1}$ and $g_{{\ell }_2}\left(\varrho \right)={(1-\varrho )}^{{\ell }_2}$ in Theorem 18 for all ${\ell }_1=1,2,\dots ,n$ and ${\ell }_2=1,2,\dots ,m$, we obtain

$$\begin{array}{c}\left|{\mathcal{T}}_2(\mathsf{\Theta };{\chi }_1,{\chi }_2)\right|\le ({\chi }_2-{\chi }_1){\left(\frac{5}{36}\right)}^{1-\frac{1}{q}}\hfill \\ \times \{{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{U_{{\ell }_1}^{\left(1\right)}}{6}-V_{{\ell }_1}^{\left(1\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{P_{{\ell }_2}^{\left(1\right)}}{6}-Q_{{\ell }_2}^{\left(1\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\\ +{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(V_{{\ell }_1}^{\left(2\right)}-\frac{U_{{\ell }_1}^{\left(2\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(Q_{{\ell }_2}^{\left(2\right)}-\frac{P_{{\ell }_2}^{\left(2\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\\ +{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(\frac{5U_{{\ell }_1}^{\left(3\right)}}{6}-V_{{\ell }_1}^{\left(3\right)}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(\frac{5P_{{\ell }_2}^{\left(3\right)}}{6}-Q_{{\ell }_2}^{\left(3\right)}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\\ \hfill +{\left[\left(\frac{1}{n}\sum _{{\ell }_1=1}^n\left(V_{{\ell }_1}^{\left(4\right)}-\frac{5U_{{\ell }_1}^{\left(4\right)}}{6}\right)\right)|{\mathsf{\Theta }}^{\prime }\left({\chi }_2\right){|}^q+\left(\frac{1}{m}\sum _{{\ell }_2=1}^m\left(Q_{{\ell }_2}^{\left(4\right)}-\frac{5P_{{\ell }_2}^{\left(4\right)}}{6}\right)\right){|{\mathsf{\Theta }}^{\prime }\left({\chi }_1\right)|}^q\right]}^{\frac{1}{q}}\},\end{array}$$

where $U_{{\ell }_1}^{\left(k\right)},V_{{\ell }_1}^{\left(k\right)},P_{{\ell }_2}^{\left(k\right)}$ and $Q_{{\ell }_2}^{\left(k\right)}$ for all $k=1,2,3,4$ are as defined in Corollary 7.

5. Conclusions

In this article, we studied algebraic properties of a new generic class of functions called the $(n,m)$–generalized convex function; based on this, we proposed $\mathtt{HH}$ inequalities. Moreover, we obtained new midpoint-type inequalities of Ostrowski and Simpson based on our new definition, using well-known integral identities. Finally, we observed that the new, defined convex function is a powerful type of function used to investigate various inequalities in the real analysis field.