(q₁,q₂)-Trapezium-Like Inequalities Involving Twice Differentiable Generalized m-Convex Functions and Applications

Abstract

A new auxiliary result pertaining to twice (${\mathrm{q}}_1,{\mathrm{q}}_2$)-differentiable functions is derived. Using this new auxiliary result, some new versions of Hermite–Hadamard’s inequality involving the class of generalized $\mathfrak{m}$-convex functions are obtained. Finally, to demonstrate the significance of the main outcomes, some applications are discussed for hypergeometric functions, Mittag–Leffler functions, and bounded functions.

1. Introduction and Preliminaries

A set $\mathrm{C}\subseteq \mathbb{R}$ is said to be convex if

$$\begin{array}{c}\hfill (1-\tilde{\varpi }){\mathfrak{y}}_3+\tilde{\varpi }{\mathfrak{y}}_4\in \mathrm{C},\forall {\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{C},\tilde{\varpi }\in [0,1].\end{array}$$

A function $\tilde{\mathfrak{B}}:\mathrm{C}\to \mathbb{R}$ is said to be convex if

$$\begin{array}{c}\hfill (1-\tilde{\varpi })\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)+\tilde{\varpi }\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\ge \tilde{\mathfrak{B}}((1-\tilde{\varpi }){\mathfrak{y}}_3+\tilde{\varpi }{\mathfrak{y}}_4),\forall {\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{C},\tilde{\varpi }\in [0,1].\end{array}$$

Hermite–Hadamard’s inequality (also known as the trapezium inequality) is a famous result pertaining to convex functions. It reads as follows:

Let $\tilde{\mathfrak{B}}:[{\mathfrak{y}}_3,{\mathfrak{y}}_4]\mapsto \mathbb{R}$ be a convex function; then,

$$\frac{\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)+\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)}{2}\ge \frac{1}{{\mathfrak{y}}_4-{\mathfrak{y}}_3}\underset{{\mathfrak{y}}_3}{\overset{{\mathfrak{y}}_4}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right)\mathrm{d}\mathfrak{x}\ge \tilde{\mathfrak{B}}\left(\frac{{\mathfrak{y}}_3+{\mathfrak{y}}_4}{2}\right),$$

Recently, a variety of different new approaches have been employed to obtain new refinements of trapezium’s inequality. For details, see [1,2]. Tariboon and Ntouyas [3] obtained a ${\mathrm{q}}_2$-analogue of trapezium’s inequality using the concepts of quantum calculus (also known as calculus without limits) on finite intervals. In quantum calculus, basically, we establish the ${\mathrm{q}}_2$-analogues of classical mathematical objects, which can be recaptured by taking ${\mathrm{q}}_2\to 1^{-}$ (for details, see [4]). Alp et al. [5] obtained a corrected ${\mathrm{q}}_2$-analogue of trapezium’s inequality. For more information, interested readers are referred to [6,7,8,9,10,11]. In recent years, the classical concepts of quantum calculus have also been extended and generalized in different directions. Chakarabarti and Jagannathan [12] introduced post-quantum calculus, which is a significant generalization of quantum calculus. In quantum calculus, we deal with ${\mathrm{q}}_2$-numbers with one base ${\mathrm{q}}_2$, while in post-quantum calculus, we deal with ${\mathrm{q}}_1$ and ${\mathrm{q}}_2$-numbers with two independent variables ${\mathrm{q}}_1$ and ${\mathrm{q}}_2$. Tunç and Gov [13] introduced the concepts of $({\mathrm{q}}_1,{\mathrm{q}}_2)$-derivatives ${}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right)$ and $({\mathrm{q}}_1,{\mathrm{q}}_2)$-integrals on finite intervals

$$\underset{{\mathfrak{y}}_3}{\overset{\mathfrak{x}}{\int }}\tilde{\mathfrak{B}}\left(\tilde{\varpi }\right){}_{{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }$$

for all $\mathfrak{x}\ne {\mathfrak{y}}_3,$ where $\mathfrak{x}\in \mathrm{K}\subset \mathbb{R}$ as follows.

Definition 1

([13]). Let $\tilde{\mathfrak{B}}:\mathrm{K}\subset \mathbb{R}\mapsto \mathbb{R}$ be a continuous function, and let $\mathfrak{x}\in \mathrm{K},$ where $0<{\mathrm{q}}_2<{\mathrm{q}}_1\le 1$. Then, the $({\mathrm{q}}_1,{\mathrm{q}}_2)$-derivative on K of function $\tilde{\mathfrak{B}}$ at $\mathfrak{x}$ is defined as

$$\begin{array}{c}\hfill {}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right)=\frac{\tilde{\mathfrak{B}}({\mathrm{q}}_1\mathfrak{x}+(1-{\mathrm{q}}_1){\mathfrak{y}}_3)-\tilde{\mathfrak{B}}({\mathrm{q}}_2\mathfrak{x}+(1-{\mathrm{q}}_2){\mathfrak{y}}_3)}{({\mathrm{q}}_1-{\mathrm{q}}_2)(\mathfrak{x}-{\mathfrak{y}}_3)},\mathfrak{x}\ne {\mathfrak{y}}_3.\end{array}$$

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Definition 2

([13]). Let $\tilde{\mathfrak{B}}:\mathrm{K}\subset \mathbb{R}\mapsto \mathbb{R}$ be a continuous function. Then, $({\mathrm{q}}_1,{\mathrm{q}}_2)$-integral on K is defined as

$$\begin{array}{c}\hfill \underset{{\mathfrak{y}}_3}{\overset{\mathfrak{x}}{\int }}\tilde{\mathfrak{B}}\left(\tilde{\varpi }\right){}_{{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }=({\mathrm{q}}_1-{\mathrm{q}}_2)(\mathfrak{x}-{\mathfrak{y}}_3)\sum _{\mathrm{n}=0}^{\infty }\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\tilde{\mathfrak{B}}\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\mathfrak{x}+\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right){\mathfrak{y}}_3\right),\end{array}$$

for $\mathfrak{x}\in \mathrm{K}$ and $\mathfrak{x}\ne {\mathfrak{y}}_3.$

Since its appearance, several new variants of classical integral inequalities have been obtained using the concepts of post-quantum calculus. For example, see [14,15,16,17,18,19,20,21].

The classical concepts of convexity have also been extended and generalized in different directions using novel and innovative ideas. For example, Cortez et al. [22] presented a new generalization of the convexity class as follows:

Definition 3

([22]). Let ${\tilde{\aleph }}_1,{\tilde{\aleph }}_2>0$ and ${\tilde{\aleph }}_3=({\tilde{\aleph }}_3\left(0\right),\dots ,{\tilde{\aleph }}_3\left(\mathrm{k}\right),\dots )$ be a bounded sequence of positive real numbers. A non-empty set $\mathrm{I}\subseteq \mathbb{R}$ is said to be generalized convex if

$$\begin{array}{c}\hfill {\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}({\mathfrak{y}}_4-{\mathfrak{y}}_3)\in \mathrm{I},\forall {\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{I},\tilde{\varpi }\in [0,1].\end{array}$$

Definition 4

([22]). Let ${\tilde{\aleph }}_1,{\tilde{\aleph }}_2>0$ and ${\tilde{\aleph }}_3=({\tilde{\aleph }}_3\left(0\right),\dots ,{\tilde{\aleph }}_3\left(\mathrm{k}\right),\dots )$ be a bounded sequence of positive real numbers. A function $\tilde{\mathfrak{B}}:\mathrm{I}\subseteq \mathbb{R}\mapsto \mathbb{R}$ is said to be generalized convex if

$$\begin{array}{c}\hfill \tilde{\mathfrak{B}}({\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}({\mathfrak{y}}_4-{\mathfrak{y}}_3))\le (1-\tilde{\varpi })\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)+\tilde{\varpi }\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right),\forall {\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{I},\tilde{\varpi }\in [0,1].\end{array}$$

Here, ${\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(z\right)$ is Raina’s function introduced and studied by [23], as follows:

$$\begin{array}{c}\hfill {\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(z\right)={\mathfrak{R}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3\left(0\right),{\tilde{\aleph }}_3\left(1\right),\dots }\left(z\right):=\sum _{\mathrm{k}=0}^{\infty }\frac{{\tilde{\aleph }}_3\left(\mathrm{k}\right)}{\Gamma ({\tilde{\aleph }}_1\mathrm{k}+{\tilde{\aleph }}_2)}{z}^{\mathrm{k}},z\in \mathbb{C},\end{array}$$

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where ${\tilde{\aleph }}_1,{\tilde{\aleph }}_2>0$, with bounded modulus $\left|z\right|<M$, and ${\tilde{\aleph }}_3=\{{\tilde{\aleph }}_3\left(0\right),{\tilde{\aleph }}_3\left(1\right),\dots ,{\tilde{\aleph }}_3\left(\mathrm{k}\right),\dots \}$ is a bounded sequence of positive real numbers. For more details, see [23].

Taking inspiration from the above definition, we now define the class of generalized $\mathfrak{m}$-convex sets.

Definition 5.

A set $\mathrm{K}\subset \mathbb{R}$ is said to be generalized $\mathfrak{m}$-convex if $\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\in \mathrm{K}$ for every ${\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{K}$ and $\mathfrak{m}\in (0,1],\tilde{\varpi }\in [0,1].$

Now, we introduce the following class of generalized $\mathfrak{m}$-convex functions.

Definition 6.

A function $\tilde{\mathfrak{B}}:\mathrm{K}\mapsto \mathbb{R}$ is said to be generalized $\mathfrak{m}$-convex if

$$\begin{array}{c}\hfill \tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\le \mathfrak{m}(1-\tilde{\varpi })\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)+\tilde{\varpi }\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right),\end{array}$$

$\forall {\mathfrak{y}}_3,{\mathfrak{y}}_4\in \mathrm{K},\mathfrak{m}\in (0,1],\tilde{\varpi }\in [0,1].$

Remark 1.

Note the following:

• I.If we take ${\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)={\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3$ in Definition 6, then we have the definition of an $\mathfrak{m}$-convex function; see [24].
• II.If we choose $\mathfrak{m}=1$ in Definition 6, then we have Definition 4.

The main motivation behind this paper is to derive a new post-quantum integral identity involving twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable functions. Using this new identity as an auxiliary result, we obtain some new variants of Hermite–Hadamard’s inequality, essentially via the class of generalized $\mathfrak{m}$-convex functions. In order to support our theoretical results, we also offer some applications to hypergeometric functions, Mittag–Leffler functions, and bounded functions. We hope that the ideas and techniques in this paper will inspire interested readers working in this field.

2. Main Results

In this section, we derive a new post-quantum integral identity. This result will be helpful in obtaining the main results of this paper. First, let us denote

$$R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{\epsilon {\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right):={\left[{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right]}^{\epsilon }.$$

Lemma 1.

Let $\tilde{\mathfrak{B}}:\mathrm{I}\mapsto \mathbb{R}$ be a twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable function on ${\mathrm{I}}^{\circ }$ (the interior of I) and ${}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}$ be continuous on $\mathrm{I},$ where $0<{\mathrm{q}}_2<{\mathrm{q}}_1\le 1.$ Then,

$$\begin{array}{cc}\hfill & \overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\hfill \\ \hfill & =\frac{{\mathrm{q}}_2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)+{\mathrm{q}}_1\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\hfill \\ \hfill & -\frac{1}{{{\mathrm{q}}_1}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\underset{\mathfrak{m}{\mathfrak{y}}_3}{\overset{\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\mathfrak{x}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }.\hfill \end{array}$$

Proof. 

Applying Definition 1, we have

$$\begin{array}{cc}\hfill & {}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\hfill \\ \hfill & ={}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\left({}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\right)\hfill \\ \hfill & =\frac{{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{\tilde{\varpi }({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & =\frac{1}{\tilde{\varpi }({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\left[\frac{\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\mathrm{q}}_2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{\tilde{\varpi }p({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right.\hfill \\ \hfill & \left.-\frac{\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\mathrm{q}}_2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{\tilde{\varpi }q({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right]\hfill \\ \hfill & {\mathrm{q}}_2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\hfill \\ \hfill & =\frac{-({\mathrm{q}}_1+{\mathrm{q}}_2)\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\mathrm{q}}_2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)+{\mathrm{q}}_1\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{\mathrm{q}}_2{\tilde{\varpi }}^2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}.\hfill \end{array}$$

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Now using Definition 2, we obtain

$$\begin{array}{cc}\hfill & \underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & =\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })\hfill \\ \hfill & \times \frac{{\mathrm{q}}_2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-({\mathrm{q}}_1+{\mathrm{q}}_2)\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)+{\mathrm{q}}_1\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2^2\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\tilde{\varpi }}^2{\mathrm{q}}_1{\mathrm{q}}_2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & =\frac{1}{{\mathrm{q}}_1{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & \times \left[{\mathrm{q}}_2{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-({\mathrm{q}}_1+{\mathrm{q}}_2){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\right.\hfill \\ \hfill & \left.+{\mathrm{q}}_1{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\right]\hfill \\ \hfill & -{\mathrm{q}}_2\left\{\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{\mathrm{q}}_2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right.\hfill \\ \hfill & -\frac{({\mathrm{q}}_1+{\mathrm{q}}_2)({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & \left.+\frac{{\mathrm{q}}_1({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{{\mathrm{q}}_2}^3{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right\}\hfill \\ \hfill & =\frac{{\mathrm{q}}_2\left[{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\right]}{{\mathrm{q}}_1{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & -\frac{{\mathrm{q}}_1\left[{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)-{\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\right]}{{\mathrm{q}}_1{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & -{\mathrm{q}}_2\left\{\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{\mathrm{q}}_2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right.\hfill \\ \hfill & -\frac{{{\mathrm{q}}_1}^2({\mathrm{q}}_1+{\mathrm{q}}_2)({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+2}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+2}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & \left.+\frac{{{\mathrm{q}}_1}^3({\mathrm{q}}_1-{\mathrm{q}}_2){\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right){\displaystyle \sum _{\mathrm{n}=0}^{\infty }}\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+3}}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{{\mathrm{q}}_1}^2\frac{{{\mathrm{q}}_2}^{\mathrm{n}+2}}{{{\mathrm{q}}_1}^{\mathrm{n}+3}}{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1{{\mathrm{q}}_2}^3{({\mathrm{q}}_1-{\mathrm{q}}_2)}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\right\}\hfill \\ \hfill & =\frac{{\mathrm{q}}_2\left[\tilde{\mathfrak{B}}(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)-\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)\right]-{\mathrm{q}}_1\left[\tilde{\mathfrak{B}}(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right))-\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)\right]}{{\mathrm{q}}_1{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & -\frac{{\mathrm{q}}_1+{\mathrm{q}}_2}{{{\mathrm{q}}_1}^3{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\underset{\mathfrak{m}{\mathfrak{y}}_3}{\overset{\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }-\frac{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1}{{\mathrm{q}}_1{{\mathrm{q}}_2}^2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)\hfill \\ \hfill & +\frac{\tilde{\mathfrak{B}}(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right))}{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)R_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\hfill \\ \hfill & =\frac{\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1{\mathrm{q}}_2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}+\frac{\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}-\frac{{\mathrm{q}}_1+{\mathrm{q}}_2}{{{\mathrm{q}}_1}^3{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{3{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\underset{\mathfrak{m}{\mathfrak{y}}_3}{\overset{\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\mathfrak{x}.\hfill \end{array}$$

Multiplying both sides of the above equality by $\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}$, we acquire the required result. □

Using Lemma 1, we can obtain the following new results.

Theorem 1.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}|$ is a generalized $\mathfrak{m}$-convex function, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)(\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)|+{{\mathrm{q}}_1}^3|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\left|\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^2({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}.\hfill \end{array}$$

Proof. 

Using Lemma 1, the property of the modulus and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)|\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-\tilde{\varpi })(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^2(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)(\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)|+{{\mathrm{q}}_1}^3|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^2({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}.\hfill \end{array}$$

This completes the proof. □

Theorem 2.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\mathfrak{m}{\mathrm{D}}_1|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right){|}^r+{\mathrm{d}}_2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_1:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\left(\frac{{{\mathrm{q}}_2}^{2\mathrm{n}}}{{{\mathrm{q}}_1}^{2\mathrm{n}+2}}-\frac{{{\mathrm{q}}_2}^{3\mathrm{n}}}{{{\mathrm{q}}_1}^{3\mathrm{n}+3}}\right){\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r,\end{array}$$

(4)

$$\begin{array}{c}\hfill {\mathrm{d}}_2:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\frac{{{\mathrm{q}}_2}^{3\mathrm{n}}}{{{\mathrm{q}}_1}^{3\mathrm{n}+3}}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r.\end{array}$$

(5)

Proof. 

Using Lemma 1, the power mean inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\frac{1}{{\mathrm{q}}_1+{\mathrm{q}}_2}\right)}^{1-\frac{1}{r}}\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-\tilde{\varpi }){(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right.\hfill \\ \hfill & {\left.+|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^2{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\mathfrak{m}{\mathrm{D}}_1|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 3.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{{\mathrm{d}}_3}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1-{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+({\mathrm{q}}_1+{\mathrm{q}}_2){\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_3:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\frac{{{\mathrm{q}}_2}^{2\mathrm{n}}}{{{\mathrm{q}}_1}^{2\mathrm{n}+2}}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^s.\end{array}$$

(6)

Proof. 

Using Lemma 1, Hölder’s inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}\hfill \\ \hfill & \times {\left(\mathfrak{m}|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^2{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{{\mathrm{d}}_3}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1-{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+({\mathrm{q}}_1+{\mathrm{q}}_2){\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 4.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\mathfrak{m}{\mathrm{D}}_4|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_5{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_4:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }{\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^{r+1}\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right){\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r\end{array}$$

(7)

and

$$\begin{array}{c}\hfill {\mathrm{d}}_5:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }{\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^{r+3}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r.\end{array}$$

(8)

Proof. 

Using Lemma 1, the power mean inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}1{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r(1-\tilde{\varpi }){(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right.\hfill \\ \hfill & {\left.+|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^{r+2}{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\mathfrak{m}{\mathrm{D}}_4|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{d}}_5{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 5.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_6^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({\mathrm{q}}_1+{\mathrm{q}}_2-1)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_6:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }{\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^{s+1}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^s.\end{array}$$

(9)

Proof. 

Using Lemma 1, Hölder’s inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^s{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\hfill \\ \hfill & \times {\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^s{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_6^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({\mathrm{q}}_1+{\mathrm{q}}_2-1)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 6.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}{\left(\mathfrak{m}{\mathrm{d}}_7|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{d}}_8{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_7:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}-\frac{{{\mathrm{q}}_2}^{2\mathrm{n}}}{{{\mathrm{q}}_1}^{2\mathrm{n}+2}}\right){\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r\end{array}$$

(10)

and

$$\begin{array}{c}\hfill {\mathrm{d}}_8:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\frac{{{\mathrm{q}}_2}^{2\mathrm{n}}}{{{\mathrm{q}}_1}^{2\mathrm{n}+2}}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^r.\end{array}$$

(11)

Proof. 

Using Lemma 1, Hölder’s inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tilde{\varpi }){(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right.\hfill \\ \hfill & {\left.+|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right){|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }{(1-{\mathrm{q}}_2\tilde{\varpi })}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}{\left(\mathfrak{m}{\mathrm{D}}_7|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_8{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 7.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_9^{\frac{1}{s}}{\left(\mathfrak{m}\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_9:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^s.\end{array}$$

(12)

Proof. 

Using Lemma 1, Hölder’s inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r(1-\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right.\hfill \\ \hfill & {\left.+|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^{r+1}{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_9^{\frac{1}{s}}{\left(\mathfrak{m}\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 8.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\frac{{{\mathrm{q}}_1}^2}{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^3{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Proof. 

Using Lemma 1, the power mean inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}\hfill \\ \hfill & \times {\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-\tilde{\varpi })(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^2(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\frac{{{\mathrm{q}}_1}^2}{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^3{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 9.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}\left(\mathfrak{m}\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{({\mathrm{q}}_1-{\mathrm{q}}_2)(1+{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}+\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)\right|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}-\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Proof. 

Using Lemma 1, the power mean inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r(1-{\mathrm{q}}_2\tilde{\varpi }){|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{1-\frac{1}{r}}\hfill \\ \hfill & \times {\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^r(1-\tilde{\varpi })(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^{r+1}(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}\left(\mathfrak{m}\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{({\mathrm{q}}_1-{\mathrm{q}}_2)(1+{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}+\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)\right|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}-\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof. □

Theorem 10.

Under the assumptions of Lemma 1, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{D}}_{10}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^3-{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where

$$\begin{array}{c}\hfill {\mathrm{d}}_{10}:=({\mathrm{q}}_1-{\mathrm{q}}_2)\sum _{\mathrm{n}=0}^{\infty }{\left(\frac{{{\mathrm{q}}_2}^{\mathrm{n}}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^{s+1}{\left(1-\frac{{{\mathrm{q}}_2}^{\mathrm{n}+1}}{{{\mathrm{q}}_1}^{\mathrm{n}+1}}\right)}^s.\end{array}$$

(13)

Proof. 

Using Lemma 1, Hölder’s inequality, and the given hypothesis, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi })|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^s(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}{\left(\underset{0}{\overset{1}{\int }}(1-{\mathrm{q}}_2\tilde{\varpi }){|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)|}^r{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}{\left(\underset{0}{\overset{1}{\int }}{\tilde{\varpi }}^s{(1-{\mathrm{q}}_2\tilde{\varpi })}^s{}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{s}}\hfill \\ \hfill & \times {\left(\mathfrak{m}|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r\underset{0}{\overset{1}{\int }}(1-\tilde{\varpi })(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }+{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }\right)}^{\frac{1}{r}}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{R}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_{10}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^3-{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

This completes the proof □

3. Applications

In this section, we discuss some applications of our main results.

3.1. Applications to Hypergeometric Functions

Now, we derive some other inequalities involving hypergeometric functions and Mittag–Leffler functions.

From the relation in (1), if we set ${\tilde{\aleph }}_1=1,{\tilde{\aleph }}_2=0$ and ${\tilde{\aleph }}_3\left(\mathrm{k}\right)=\frac{{\left(\mathsf{\varphi }\right)}_{\mathrm{k}}{\left(\mathsf{\psi }\right)}_{\mathrm{k}}}{{\left(\mathsf{\eta }\right)}_{\mathrm{k}}}\ne 0$, where $\mathsf{\varphi },\mathsf{\psi }$, and $\mathsf{\eta }$ are parameters that may be real or complex values; ${\left(\mathfrak{m}\right)}_{\mathrm{k}}$ is defined as ${\left(\mathfrak{m}\right)}_{\mathrm{k}}=\frac{\Gamma (\mathfrak{m}+\mathrm{k})}{\Gamma \left(\mathfrak{m}\right)}$, and its domain is restricted as $\left|\mathfrak{x}\right|\le 1$, then we have the following hypergeometric function:

$$\begin{array}{c}\hfill {\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta },\mathfrak{x}):=\sum _{\mathrm{k}=0}^{\infty }\frac{{\left(\mathsf{\varphi }\right)}_{\mathrm{k}}{\left(\mathsf{\psi }\right)}_{\mathrm{k}}}{\mathrm{k}!{\left(\mathsf{\eta }\right)}_{\mathrm{k}}}{\mathfrak{x}}^{\mathrm{k}}.\end{array}$$

Let us denote

$${\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right):={\left[{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right]}^2.$$

Lemma 2.

Let $\tilde{\mathfrak{B}}:\mathrm{I}\mapsto \mathbb{R}$ be a twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable function on ${\mathrm{I}}^{\circ }$ (the interior of I) and ${}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}$ be continuous and $\mathrm{I},$ where $0<{\mathrm{q}}_2<{\mathrm{q}}_1\le 1.$ Then,

$$\begin{array}{cc}\hfill & {\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\hfill \\ \hfill & =\frac{{\mathrm{q}}_2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)+{\mathrm{q}}_1\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\hfill \\ \hfill & -\frac{1}{{{\mathrm{q}}_1}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\underset{\mathfrak{m}{\mathfrak{y}}_3}{\overset{\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\mathfrak{x}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }.\hfill \end{array}$$

Theorem 11.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}|$ is a generalized $\mathfrak{m}$-convex function, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)(\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)|+{{\mathrm{q}}_1}^3|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^2({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}.\hfill \end{array}$$

Theorem 12.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\mathfrak{m}{\mathrm{D}}_1|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ are given by (4) and (5), respectively.

Theorem 13.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{{\mathrm{d}}_3}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1-{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+({\mathrm{q}}_1+{\mathrm{q}}_2){\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_3$ is given by (6).

Theorem 14.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\mathfrak{m}{\mathrm{D}}_4|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_5{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_4$ and ${\mathrm{d}}_5$ are given by (7) and (8).

Theorem 15.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_6^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({\mathrm{q}}_1+{\mathrm{q}}_2-1)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_6$ is given by (9).

Theorem 16.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}{\left(\mathfrak{m}{\mathrm{d}}_7|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{d}}_8{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_7$ and ${\mathrm{d}}_8$ are given by (10) and (11).

Theorem 17.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_9^{\frac{1}{s}}\hfill \\ \hfill & \times {\left(\mathfrak{m}\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}{|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_9$ is given by (12).

Theorem 18.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\frac{{{\mathrm{q}}_1}^2}{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2}\right)}^{1-\frac{1}{r}}\hfill \\ \hfill & \times {\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^3{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Theorem 19.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}\left(\mathfrak{m}\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{({\mathrm{q}}_1-{\mathrm{q}}_2)(1+{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}+\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)\right|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}-\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Theorem 20.

Under the assumptions of Lemma 2, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$, then

$$\begin{array}{cc}\hfill & \left|{\overline{\Xi }}_1(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left(\mathsf{\varphi },\mathsf{\psi };\mathsf{\eta };{\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_{10}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^3-{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_{10}$ is given by (13).

3.2. Applications to Mittag–Leffler Functions

Moreover, if we take ${\tilde{\aleph }}_3=(1,1,1,\dots ),{\tilde{\aleph }}_2=1$, and ${\tilde{\aleph }}_1=\mathsf{\varphi }$ with $\Re \left(\mathsf{\varphi }\right)>0$, then we obtain the well-known Mittag–Leffler function:

$$\begin{array}{c}\hfill {\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left(\mathfrak{x}\right):=\sum _{\mathrm{k}=0}^{\infty }\frac{1}{\Gamma (1+\mathsf{\varphi }\mathrm{k})}{\mathfrak{x}}^{\mathrm{k}}.\end{array}$$

Let us denote

$${\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right):={\left[{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right]}^2.$$

Lemma 3.

Let $\tilde{\mathfrak{B}}:\mathrm{I}\mapsto \mathbb{R}$ be a twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable function on ${\mathrm{I}}^{\circ }$ (the interior ofI) and ${}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}$ be continuous and $\mathrm{I},$ where $0<{\mathrm{q}}_2<{\mathrm{q}}_1\le 1.$ Then,

$$\begin{array}{cc}\hfill & \overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\hfill \\ \hfill & =\frac{{\mathrm{q}}_2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right)+{\mathrm{q}}_1\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\hfill \\ \hfill & -\frac{1}{{{\mathrm{q}}_1}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}\underset{\mathfrak{m}{\mathfrak{y}}_3}{\overset{\mathfrak{m}{\mathfrak{y}}_3+{\mathrm{q}}_1^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{\int }}\tilde{\mathfrak{B}}\left(\mathfrak{x}\right){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\mathfrak{x}\hfill \\ \hfill & =\frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{\mathrm{q}}_1+{\mathrm{q}}_2}\underset{0}{\overset{1}{\int }}\tilde{\varpi }(1-{\mathrm{q}}_2\tilde{\varpi }){}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3+\mathfrak{m}\tilde{\varpi }{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)\right){}_0{\mathrm{d}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}\tilde{\varpi }.\hfill \end{array}$$

Theorem 21.

Under the assumptions of Lemma 3, if $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}|$ is a generalized $\mathfrak{m}$-convex function, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)(\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)|+{{\mathrm{q}}_1}^3|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\left|\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^2({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}.\hfill \end{array}$$

Theorem 22.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$, then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\mathfrak{m}{\mathrm{d}}_1|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left(\mathfrak{m}{\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_1$ and ${\mathrm{d}}_2$ are given by (4) and (5), respectively.

Theorem 23.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{{\mathrm{d}}_3}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1-{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+({\mathrm{q}}_1+{\mathrm{q}}_2){\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_3$ is given by (6).

Theorem 24.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$. Then

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\mathfrak{m}{\mathrm{d}}_4|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{D}}_5{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_4$ and ${\mathrm{d}}_5$ are given by (7) and (8).

Theorem 25.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_6^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({\mathrm{q}}_1+{\mathrm{q}}_2-1)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

(14)

where ${\mathrm{d}}_6$ is given by (9).

Theorem 26.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}{\left(\mathfrak{m}{\mathrm{d}}_7|{}_{{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{\mathrm{d}}_8{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_7$ and ${\mathrm{d}}_8$ are given by (10) and (11).

Theorem 27.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\mathfrak{x}\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_9^{\frac{1}{s}}{\left(\mathfrak{m}\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_9$ is given by (12).

Theorem 28.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\frac{{{\mathrm{q}}_1}^2}{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^3{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Theorem 29.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r\ge 1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}\left(\mathfrak{m}\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{({\mathrm{q}}_1-{\mathrm{q}}_2)(1+{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}+\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right)\right|}^r\right.\hfill \\ \hfill & {\left.+\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}-\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r\right)}^{\frac{1}{r}}.\hfill \end{array}$$

Theorem 30.

Under the assumptions of Lemma 3, $|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}{|}^r$ is a generalized $\mathfrak{m}$-convex function for $r>1$ and $\frac{1}{s}+\frac{1}{r}=1$. Then,

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{\mathsf{\varphi }}^2\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_{10}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^3-{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2{\mathrm{q}}_2)|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_3\right){|}^r+{{\mathrm{q}}_1}^2{\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\left({\mathfrak{y}}_4\right)\right|}^r}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}},\hfill \end{array}$$

where ${\mathrm{d}}_{10}$ is given by (13).

3.3. Applications to Bounded Functions

We suppose that the following condition is satisfied:

$$\left|{}_{\mathfrak{m}{\mathfrak{y}}_3}{\mathrm{D}}_{{\mathrm{q}}_1,{\mathrm{q}}_2}^2\tilde{\mathfrak{B}}\right|\le \mathcal{M},$$

which means that the twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable function $\tilde{\mathfrak{B}}$ is in an absolute value bounded from the positive real number $\mathcal{M}$.

Proposition 1.

Under the conditions of Theorem 1, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)(\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)+{{\mathrm{q}}_1}^3)\mathcal{M}}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^2({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}.\hfill \end{array}$$

Proposition 2.

Under the conditions of Theorem 2, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\mathfrak{m}{\mathrm{d}}_1+{\mathrm{d}}_2\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 3.

Under the conditions of Theorem 3, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{{\mathrm{d}}_3}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{\mathrm{q}}_2-{\mathrm{q}}_1-{\mathrm{q}}_2)+{\mathrm{q}}_1+{\mathrm{q}}_2}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 4.

Under the conditions of Theorem 4, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\mathfrak{m}{\mathrm{d}}_4+{\mathrm{d}}_5\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 5.

Under the conditions of Theorem 5, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_6^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({\mathrm{q}}_1+{\mathrm{q}}_2-1)+1}{({\mathrm{q}}_1+{\mathrm{q}}_2)}\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 6.

Under the conditions of Theorem 6, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{s+1}-{{\mathrm{q}}_2}^{s+1}}\right)}^{\frac{1}{s}}{\left(\mathfrak{m}{\mathrm{d}}_7+{\mathrm{d}}_8\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 7.

Under the conditions of Theorem 7, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_9^{\frac{1}{s}}{\left(\mathfrak{m}\left(\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)+\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 8.

Under the conditions of Theorem 8, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}{\left(\frac{{{\mathrm{q}}_1}^2}{{{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2}\right)}^{1-\frac{1}{r}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^4-{{\mathrm{q}}_1}^3+{{\mathrm{q}}_1}^2{{\mathrm{q}}_2}^2)+{{\mathrm{q}}_1}^3}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_1}^2+{{\mathrm{q}}_2}^2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 9.

Under the conditions of Theorem 9, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{{\mathrm{q}}_1}^{2-\frac{1}{r}}{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{{({\mathrm{q}}_1+{\mathrm{q}}_2)}^{2-\frac{1}{r}}}\left(\mathfrak{m}\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+1}-{{\mathrm{q}}_2}^{r+1}}-\frac{({\mathrm{q}}_1-{\mathrm{q}}_2)(1+{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}+\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]\right.\hfill \\ \hfill & {\left.+\left[\frac{{\mathrm{q}}_1-{\mathrm{q}}_2}{{{\mathrm{q}}_1}^{r+2}-{{\mathrm{q}}_2}^{r+2}}-\frac{{\mathrm{q}}_2({\mathrm{q}}_1-{\mathrm{q}}_2)}{{{\mathrm{q}}_1}^{r+3}-{{\mathrm{q}}_2}^{r+3}}\right]\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Proposition 10.

Under the conditions of Theorem 10, we have

$$\begin{array}{cc}\hfill & \left|\overline{\Xi }(\tilde{\mathfrak{B}};{\mathrm{q}}_1,{\mathrm{q}}_2;{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{{\tilde{\aleph }}_3})\right|\hfill \\ \hfill & \le \frac{{\mathrm{q}}_1{{\mathrm{q}}_2}^2{\overline{\tilde{\mathfrak{R}}}}_{{\tilde{\aleph }}_1,{\tilde{\aleph }}_2}^{2{\tilde{\aleph }}_3}\left({\mathfrak{y}}_4-\mathfrak{m}{\mathfrak{y}}_3\right)}{({\mathrm{q}}_1+{\mathrm{q}}_2)}{\mathrm{d}}_{10}^{\frac{1}{s}}{\left(\frac{\mathfrak{m}({{\mathrm{q}}_1}^3-{{\mathrm{q}}_1}^2+{\mathrm{q}}_1{{\mathrm{q}}_2}^2+{{\mathrm{q}}_1}^2{\mathrm{q}}_2)+{{\mathrm{q}}_1}^2}{({\mathrm{q}}_1+{\mathrm{q}}_2)({{\mathrm{q}}_2}^2+{\mathrm{q}}_1{\mathrm{q}}_2+{{\mathrm{q}}_1}^2)}\right)}^{\frac{1}{r}}\mathcal{M}.\hfill \end{array}$$

Remark 2.

Interested readers can derive several new inequalities from our main results using special means of positive real numbers. We omit their proofs here.

4. Conclusions

We established a new integral identity utilizing twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable functions. Some new variants of Hermite–Hadamard’s inequality essentially involving the class of generalized $\mathfrak{m}$-convex functions were derived. In order to illustrate the efficiency and significance of our main outcomes, some applications regarding hypergeometric functions, Mittag–Leffler functions, and twice $({\mathrm{q}}_1,{\mathrm{q}}_2)$-differentiable functions that are bounded were discussed in detail. To the best of our knowledge, these results are new in the literature. Note that these $({\mathrm{q}}_1,{\mathrm{q}}_2)$-trapezium inequalities involving twice differentiable generalized $\mathfrak{m}$-convex functions are connected to theoretical methods and have applications related to fractional operators. It is also worth mentioning here that one can extend these results using fractional quantum integrals. This will be an interesting problem for future research.