New Quantum Estimates for Midpoint and Trapezoid Type Inequalities Through (α,m)-Convex Functions with Applications

The main goal of current investigation is to present two new q-integral identities for midpoint and trapezoid type inequalities. Then using these identities, we develop several new quantum estimates for midpoint and trapezoid type inequalities via (α, m)-convexity. Some special cases of these new inequalities can be turned into quantum midpoint and quantum trapezoid type inequalities for convex functions, classical midpoint and trapezoid type inequalities for convex functions without having to prove each one separately. Finally, we discuss how the special means can be used to address newly discovered inequalities.

2010 Mathematics Subject Classification. 26D10, 26D15, 26B25.

Introduction

It is well known that modern investigation, directly or indirectly, involves the applications of convexity. Due to its use and significant importance, the concept of convex sets and hence convex functions is largely generalized in various directions. The concept of convexity and its variant forms have played a fundamental role in the development of different fields. Convex functions are powerful tools for proving a large class of inequalities. Today the study of convex functions evolved into a broader theory of functions including quasi-convex functions [1-3], log convex functions [4], co-ordinated convex functions [5,6], harmonically convex functions [7], GA-convex functions [8,9], (α, m) -convex functions [10]. Convexity naturally gives rise to inequalities, Hermite-Hadamard inequalities is the fisrst consequence of convex functions. A function $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$ , where $\Im$ is an interval in $ℝ$ is called convex, if it satisfies the inequality

$\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathfrak{r}+\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{s}\mathrm{<mo\; stretchy="false">)</mo>}\le \mathfrak{w}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{s}\mathrm{<mo\; stretchy="false">)</mo>}$

where $\mathfrak{r}\mathrm{,}\mathfrak{s}\in \Im$ and $\mathfrak{w}\in \mathrm{0,1<mo\; stretchy="false">]</mo>}$ .

A class of $\left(\alpha \mathrm{,}m\right)$ -convex functions was introduced by Mihesan and stated as:

Definition 1 [10] A function $\mathfrak{F}\mathrm{<mo>:</mo>}\left[\mathrm{0,}{\mathfrak{y}}_2\right)\to ℝ$ is called (a, m)-convex, if the inequality

$\mathfrak{F}\left(\mathfrak{w}\mathfrak{r}+m\left(1-\mathfrak{w}\right)\mathfrak{s}\right)\le {\mathfrak{w}}^{\alpha }\mathfrak{F}\left(\mathfrak{r}\right)+m\left(1-{\mathfrak{w}}^{\alpha }\right)\mathfrak{F}\left(\mathfrak{s}\right)$

holds for all $\mathfrak{r}\mathrm{,}\mathfrak{s}\in \left[\mathrm{0,}{\mathfrak{y}}_2\right)$ , $\mathfrak{w}\in \left[\mathrm{0,1}\right]\mathrm{,}\alpha \in \left[\mathrm{0,1}\right]$ and $m\in \left[\mathrm{0,1}\right]\mathrm{.}$ .

It is also well known that $\mathfrak{F}$ is convex if and only if it satisfies the Hermite-Hadamard's inequality, stated below:

$\mathfrak{F}\left(\frac{{\mathfrak{y}}_1+{\mathfrak{y}}_2}{2}\right)\le \frac{1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}{\displaystyle {\int }_{{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}d\mathfrak{r}\le \frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}+\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}}{2}$

where $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$ is a convex function and ${\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\in \Im$ with ${\mathfrak{y}}_1\mathrm{<mo><</mo>}{\mathfrak{y}}_2$ Convexity is mixed with other mathematical concepts like; optimization [11], time scale [12,13], quantum and post quantum calculus [14].

On the other hand, several works in the field of q-analysis are being carried out, beginning with Euler, in order to achieve mastery in the mathematics that drives quantum computing. Q-calculus is the connection between physics and mathematics. It has a wide range of applications in many fields, e.g., mathematics, including number theory, combinatorics, orthogonal polynomials, basic hypergeometric functions, and other disciplines, as well as mechanics, theory of relativity, and quantum theory [15,16]. q-calculus also has many applications in quantum information theory, which is an interdisciplinary area that surrounds computer science, information theory, philosophy, and cryptography, among other areas [17, 18]. Euler is the inventor of this significant branch of mathematics. Newton used the q -parameter in his work on infinite series. The q -calculus that is known without limits calculus was presented by Jackson [19] in a systematic manner. In 1966, Al-Salam [20] introduced a q -analogue of the q -fractional integrals and q -Riemann-Liouville fractional. Since then, realted research has been increasing gently. In particular, in 2013, Tariboon and Ntouyas introduced the left quantum difference operator and left quantum integral in [21]. In 2020, Bermudo et al. introduced the notion of right quantum derivative and right quantum integral in [22].

Many integrals have also been investigated using quantum and post quantum calculus for different types of functions. For example, in [14,22-30], the authors proved Hermite-Hadamard integral inequalities and their left-right estimates for convex and co-ordinated convex functions by using the quantum derivative and integrals. In [31], the generalized version of q-integral inequalities was presented by Noor et al. In [32] Nwaeze et al. proved certain partametrized quantum integral inequalities for generalized quasi convex functions. Khan et al.proved Hermite-Hadamard inequality using the green function in [33]. For convex and co-ordinated convex functions, Budak et al. [34], Ali et al. [35,36] and Vivas-Cortez et al. [37] developed new quantum Simpson's and Newton's type inequalities.For quantum Ostrowski's type inequalities for convex and co-ordinated convex functions, please refer to [38-40].

Inspired by the ongoing studies, we derive some new inequalities of Midpoint and Trapezoid type inequalities for $\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{,}m\mathrm{<mo\; stretchy="false">)</mo>}$ -convex functions by utilizing quantum calculus. The fundamental benefit of these inequalities can be turned into quantum Midpoint and trapezoid type inequalities for convex functions [14,41], classical Midpoint for convex functions [42] and the classical Trapezoid type inequalities for convex functions [43] without having to prove each one separately.

This paper is summarized as follows: Section 2 provides a brief overview of the fundamentals of q-calculus as well as other related studies in this field. In Section 3, we establish two pivotal identities that play a major role in establishing the main outcomes of this paper. The Midpoint and Trapezoid type inequalities for q-differentiable functions via (a, m) -convexity are presented in section 4 and section 5. The special means are described in section 6. The connection between the findings reported here and similar findings in the literature are also taken into account. Section 7 concludes with some suggestions for future research.

Preliminaries and definitions of q-calculus

 In this section, we first present the definitions and some properties of quantum integrals. We also mention some well known inequalities for quantum integrals. Throughout this paper, let  0<q<1 be a constant.

The q-number or q -analogue of $n\in \mathbb{N}$ is given by

$n{\mathrm{<mo\; stretchy="false">]</mo>}}_q\mathrm{<mo>=</mo>}\frac{1-q^n}{1-q}\mathrm{<mo>=</mo>1}+q+q^2+\mathrm{...}+q^{n-1}\mathrm{.}$

Jackson derived the q -Jackson integral in [44] from 0 to ${\mathfrak{y}}_2$  as follows:

${\displaystyle {\int }_0^{{\mathfrak{y}}_2}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{r}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_2{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^n\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{q}^n\mathrm{<mo\; stretchy="false">)</mo>}$

provided the sum converges absolutely.

The q Jackson integral in a generic interval $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ was given by in [19] and defined as follows:

${\displaystyle {\int }_{{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{r}\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{{\mathfrak{y}}_2}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{r}-{\displaystyle {\int }_0^{{\mathfrak{y}}_1}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{r}\mathrm{.}$

Definition 2 [21] Let $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$  be a continuous function and let $\mathfrak{r}\in \Im$ . Then the $q_{{\mathfrak{y}}_1}$ derivative on $\Im$ of $\mathfrak{F}$  at $\mathfrak{r}$ is defined as

${}_{{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}-\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{r}+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}\mathfrak{r}-{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{r}\ne {\mathfrak{y}}_1\mathrm{,}\text{ (1)}$

${}_{{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}li{m}_{\mathfrak{r}\to {\mathfrak{y}}_1}{}_{{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>.}$

Definition 3 If ${\mathfrak{y}}_1\mathrm{<mo>=</mo>0}$  in (1), then we get classical q -derivative of $\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}$  at $\mathfrak{r}\in \Im$ , given by

${}_0{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}-\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{r}}\mathrm{.}$

Definition 4 [22] Let $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$  be a continuous function and let $\mathfrak{r}\in \Im$ . Then the $q^{{\mathfrak{y}}_2}$  derivative on $\Im$  of $\mathfrak{F}$  at $\mathfrak{r}$  is defined as

${}^{{\mathfrak{y}}_2}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}-\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{r}+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}\mathfrak{r}-{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{r}\ne {\mathfrak{y}}_2\mathrm{,}$

${}^{{\mathfrak{y}}_2}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}li{m}_{\mathfrak{r}\to {\mathfrak{y}}_2}{}^{{\mathfrak{y}}_2}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>.}$

Definition 5 [21] Let $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$  be a continuous function. Then the $q_{{\mathfrak{y}}_1}$ -integral on $\Im$ is defined as

${\displaystyle {\int }_{{\mathfrak{y}}_1}^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}\mathfrak{r}-{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^n\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^n\mathfrak{r}+\mathrm{<mo\; stretchy="false">(</mo>1}-q^n\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\text{ (2)}$

for $\mathfrak{r}\in \Im$ . If ${\mathfrak{y}}_1\mathrm{<mo>=</mo>0}$  in (2), then

${\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_0{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{,}$

where ${\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$  is familiar q -definite integral on $\mathrm{<mo\; stretchy="false">[</mo>0,}\mathfrak{r}\mathrm{<mo\; stretchy="false">]</mo>}$  defined by the expression

${\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_0{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{r}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^n\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^n\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>.}$

Moreover, if $c\in \mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}$ , then the q -integral on $\Im$  is defined as

${\displaystyle {\int }_c^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_{{\mathfrak{y}}_1}^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}-{\displaystyle {\int }_{{\mathfrak{y}}_1}^c}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\mathrm{.}$

Definition 6 [22] Let $\mathfrak{F}\mathrm{<mo>:</mo>}\Im \to ℝ$  be a continuous function. Then the $q^{{\mathfrak{y}}_2}$ -integral on $\Im$  is defined as

${\displaystyle {\int }_{\mathfrak{r}}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}^{{\mathfrak{y}}_2}{d}_q\mathfrak{w}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^n\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^n\mathfrak{r}+\mathrm{<mo\; stretchy="false">(</mo>1}-q^n\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\text{ (3)}$

for $\mathfrak{r}\in \Im$ . If ${\mathfrak{y}}_2\mathrm{<mo>=</mo>1}$ in (3), then

${\displaystyle {\int }_{\mathfrak{r}}^1}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}^1{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_{\mathfrak{r}}^1}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{,}$

where ${\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$ is familiar q -definite integral on $\mathrm{<mo\; stretchy="false">[</mo>0,}\mathfrak{r}\mathrm{<mo\; stretchy="false">]</mo>}$ defined by the expression

${\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_0{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{\mathfrak{r}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{r}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^n\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^n\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>.}$

Moreover, if $c\in \mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}$ , then the q -integral on $\Im$ is defined as

${\displaystyle {\int }_c^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}{\displaystyle {\int }_{{\mathfrak{y}}_1}^{\mathfrak{r}}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}-{\displaystyle {\int }_{{\mathfrak{y}}_1}^c}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\mathrm{.}$

n [14], Alp et al. proved the corresponding Hermite-Hadamard inequalities for convex functions by using $q_{{\mathfrak{y}}_1}$ -integrals, as follows:

Theorem 1  [14] If $\mathfrak{F}\mathrm{<mo>:</mo>}\left[{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\right]\to ℝ$  be a convex differentiable function on $\left[{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\right]$ and 0<q<1. Then, $q_{{\mathfrak{y}}_1}$ -Hermite-Hadamard inequalities

$\mathfrak{F}\left(\frac{q{\mathfrak{y}}_1+{\mathfrak{y}}_2}{{\left[2\right]}_q}\right)\le \frac{1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{w}\right)\begin{array}{l}{}_{{\mathfrak{y}}_1}{d}_q\mathfrak{w}\hfill \end{array}\le \frac{q\mathfrak{F}\left({\mathfrak{y}}_1\right)+\mathfrak{F}\left({\mathfrak{y}}_2\right)}{{\left[2\right]}_q}\mathrm{.}\text{ (4)}$

Bermudo et al. proved the corresponding Hermite-Hadamard inequalities for convex functions by using $q^{{\mathfrak{y}}_2}$ - integrals, as follows:

Theorem 2  [22] If $\mathfrak{F}\mathrm{<mo>:</mo>}\left[{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\right]\to ℝ$  be a convex differentiable function on $\left[{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\right]$  and 0<q<1. Then, $q^{{\mathfrak{y}}_2}$ -Hermite-Hadamard inequalities

$\mathfrak{F}\left(\frac{{\mathfrak{y}}_1+q{\mathfrak{y}}_2}{{\left[2\right]}_q}\right)\le \frac{1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{w}\right)\begin{array}{l}{}^{{\mathfrak{y}}_2}{d}_q\mathfrak{w}\hfill \end{array}\le \frac{\mathfrak{F}\left({\mathfrak{y}}_1\right)+q\mathfrak{F}\left({\mathfrak{y}}_2\right)}{{\left[2\right]}_q}\mathrm{.}\text{ (5)}$

From Theorem 1 and Theorem 2, one can write the following inequalities:

Corollary 1 [22] for any convex function $\mathfrak{F}\mathrm{<mo>:</mo>}\left[{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\right]\to ℝ$  and 0<q<1, we have

$\mathfrak{F}\left(\frac{q{\mathfrak{y}}_1+{\mathfrak{y}}_2}{{\left[2\right]}_q}\right)+\mathfrak{F}\left(\frac{{\mathfrak{y}}_1+q{\mathfrak{y}}_2}{{\left[2\right]}_q}\right)\le \frac{1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}\left\{{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{r}\right)\begin{array}{l}{}_{{\mathfrak{y}}_1}{d}_q\mathfrak{r}\hfill \end{array}+{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{r}\right)\begin{array}{l}{}^{{\mathfrak{y}}_2}{d}_q\mathfrak{r}\hfill \end{array}\right\}\le \mathfrak{F}\left({\mathfrak{y}}_1\right)+\mathfrak{F}\left({\mathfrak{y}}_2\right)$

and

$\mathfrak{F}\left(\frac{{\mathfrak{y}}_1+{\mathfrak{y}}_2}{2}\right)\le \frac{1}{2\left({\mathfrak{y}}_2-{\mathfrak{y}}_1\right)}\left\{{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{r}\right)\begin{array}{l}{}_{{\mathfrak{y}}_1}{d}_q\mathfrak{r}\hfill \end{array}+{\displaystyle \underset{{\mathfrak{y}}_1}{\overset{{\mathfrak{y}}_2}{\int }}}\mathfrak{F}\left(\mathfrak{r}\right)\begin{array}{l}{}^{{\mathfrak{y}}_2}{d}_q\mathfrak{r}\hfill \end{array}\right\}\le \frac{\mathfrak{F}\left({\mathfrak{y}}_1\right)+\mathfrak{F}\left({\mathfrak{y}}_2\right)}{2}\mathrm{.}$

Theorem 3 If $\mathfrak{F}\mathrm{<mo>:</mo><mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}\to ℝ$  is a continious function and $z\in {\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then the following identities hold:

$\begin{array}{ll}\hfill & \mathrm{<mo\; stretchy="false">(</mo>}i\mathrm{<mo\; stretchy="false">)</mo>}{}_{{\mathfrak{y}}_1}{D}_q{\displaystyle {\int }_{{\mathfrak{y}}_1}^z}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{<mo>=</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>;</mo>}\hfill \\ \hfill & \mathrm{<mo\; stretchy="false">(</mo>}ii\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle {\int }_c^z}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{<mo>=</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}-\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}c\mathrm{<mo\; stretchy="false">)</mo>}\text{for}c\in \mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}z\mathrm{<mo\; stretchy="false">)</mo>.}\hfill \end{array}$

Lemma 1  [45]For continious functions $\mathfrak{F}\mathrm{,}g\mathrm{<mo>:</mo><mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}\to ℝ$ , the following equality true:

${\displaystyle {\int }_0^c}g{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}{\frac{g\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}|}_0^c-\frac{1}{{\mathfrak{y}}_2-{\mathfrak{y}}_1}{\displaystyle {\int }_0^c}{D}_{q}g\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{w}{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{.}$

Key Identities

In this section, we establish two quantum integral identities using the integration by parts method for quantum integrals to obtain the main outcomes.

Lemma 2  For a q-differentiable function $\mathfrak{F}\mathrm{<mo>:</mo><mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}\to ℝ$  with ${}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}$ is continuous and integrable on $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , the following identity holds:

$q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}]$

$\mathrm{<mo>=</mo>}\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{.}\text{ (6)}$

Proof. From fundamental properties of quantum integrals, we have

$[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}]$

$\mathrm{<mo>=</mo>}[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_0^1}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$-{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}]$

$\mathrm{<mo>=</mo>}{\Im }_1+{\Im }_2-{\Im }_3\mathrm{.}$

Using the Lemma 1, we have

${\Im }_1\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}{\mathfrak{w}\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}|}_0^{\frac{1}{{\left[2\right]}_q}}-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}q{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)$

$-\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}q{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{.}\text{ (7)}$

Similarly, we have

${\Im }_2\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^1}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}\frac{q-1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\frac{1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_0^1}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}q{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}\frac{q-1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\frac{1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$-\frac{1}{q{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^2}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}+\frac{1-q}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\text{ (8)}$

and

${\Im }_3\mathrm{<mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-\frac{1}{q}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}-\frac{1}{q{\left[2\right]}_q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\mathrm{<mo\; stretchy="false">)</mo>}+\frac{1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}q{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{.}\text{ (9)}$

Thus from (7), (8) and (9), we have

${\Im }_1+{\Im }_2-{\Im }_3\mathrm{<mo>=</mo>}\frac{1}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\mathrm{<mo\; stretchy="false">)</mo>}-\frac{1}{q{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^2}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\text{ (10)}$

and we obtain required equality (6) by multiplying $q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$ on both sides of (10). Thus, the proof is accomplished.

Remark 1 In Lemma 2, we have

• if we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1,}$ then we find [14, Lemma 11].

If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$  and later taking $q\to 1^{-}$ , then we find [42, Lemma 2.1].

Lemma 3 For a q-differentiable function $\mathfrak{F}\mathrm{<mo>:</mo><mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}\to ℝ$  with ${}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}$ is continious and integrable on $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , the following equality holds:

$\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\displaystyle {\int }_0^1}\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}{\mathrm{<mo\; stretchy="false">)</mo><mi>\; </mi>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\text{ (11)}$

Proof. From fundamental properties of quantum integral, we have

${\displaystyle {\int }_0^1}\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}{\mathrm{<mo\; stretchy="false">)</mo><mi>\; </mi>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}{\frac{\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathfrak{w}+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}|}_0^1+\frac{{\left[2\right]}_q}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_0^1}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}-\frac{q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}+\frac{{\left[2\right]}_q}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_0^1}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}q\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}-\frac{q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}+\frac{{\left[2\right]}_q\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{\displaystyle \sum _{n\mathrm{<mo>=</mo>0}}^{\infty }}{q}^{n+1}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^{n+1}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q^{n+1}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo>}-\frac{q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}+\frac{{\left[2\right]}_q\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\left({\displaystyle \sum _{k\mathrm{<mo>=</mo>0}}^{\infty }}{q}^k\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{q}^k{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q^k\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}-\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right)$

$\mathrm{<mo>=</mo>}-\frac{q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}+\frac{{\left[2\right]}_q}{q{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^2}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}-\frac{{\left[2\right]}_q\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}$

$\mathrm{<mo>=</mo>}-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}+\frac{{\left[2\right]}_q}{q{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}^2}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{.}\text{ (12)}$

and we obtain the required equality (11) by multiplying $\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}$ on both sides of (12). Thus, the proof is accomplished.

Remark 2 In Lemma 3, we have

• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [41, Lemma 3.1].

If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$  and later taking limit as $q\to 1^{-}$ , then we find [43, Lemma 2.1].

Midpoint Type inequalities for (α, m) -convex functions

Theorem 4  Under the assumption of Lemma 2, if ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo>|</mo>}$  is $\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{,}m\mathrm{<mo\; stretchy="false">)</mo>}$  convex function over $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then we have the following midpoint type inequality:

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}$

$+m\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo><mo\; stretchy="false">]</mo>,}\text{ (13)}$

where

$A_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^{\alpha +1}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q^{\alpha +2}{\left[\alpha +2\right]}_q}$

$A_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{w}\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q^3}-\frac{1}{{\left[2\right]}_q^{\alpha +2}{\left[\alpha +2\right]}_q}$

$A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\mathfrak{w}}^{\alpha }\left(\frac{1}{q}-\mathfrak{w}\right)d_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{q{\left[\alpha +1\right]}_q}-\frac{1}{{\left[\alpha +2\right]}_q}$

$-\frac{1}{q{\left[2\right]}_q^{\alpha +1}{\mathrm{<mo\; stretchy="false">[</mo>}\alpha +\mathrm{1<mo\; stretchy="false">]</mo>}}_q}+\frac{1}{{\left[2\right]}_q^{\alpha +2}{\left[\alpha +2\right]}_q}$

$A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{1}{q}-\mathfrak{w}\right)\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q^3}-\frac{1}{q{\left[\alpha +1\right]}_q}+\frac{1}{{\left[\alpha +2\right]}_q}$

$+\frac{1}{q{\left[2\right]}_q^{\alpha +1}{\mathrm{<mo\; stretchy="false">[</mo>}\alpha +\mathrm{1<mo\; stretchy="false">]</mo>}}_q}-\frac{1}{{\left[2\right]}_q^{\alpha +2}{\left[\alpha +2\right]}_q}\mathrm{.}$

Proof. By taking modulus in (6), and using (a, m) -convexity of ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo>|</mo>}$ , we have

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{w}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{1}{q}-\mathfrak{w}\right){\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}]$

$\le {\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^{\alpha +1}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}+{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}m\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-{\mathfrak{w}}^{\alpha +1}\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{{\mathfrak{w}}^{\alpha }}{q}-{\mathfrak{w}}^{\alpha +1}\right){\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}m\left(\frac{1}{q}-\mathfrak{w}\right)\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[A_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}+m{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}$

$+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}+m{A}_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}]$

$\mathrm{<mo>=</mo>}q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">[</mo><mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}+m\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo><mo\; stretchy="false">]</mo>.}$

Thus, the proof is accomplished.

Remark 3 In Theorem 4, we have

• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [14, Theorem 13].
• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$  and later taking the limit as $q\to 1^{-}$ , then we find [42, Theorem 2.2].

Theorem 5  Under the assumption of Lemma 2, If ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{,}r\ge 1$  is (a, m) convex function over $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then we have the following midpoint type inequality:

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q^{\frac{\mathrm{3<mo\; stretchy="false">(</mo>}r-\mathrm{1<mo\; stretchy="false">)</mo>}}{r}}}[{\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{r}}$

$+{\mathrm{<mo\; stretchy="false">(</mo>}{A}_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{A}_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{r}}]\text{ (14)}$

Proof. By taking modulus in (6), and using power mean inequality, we have

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathrm{<mo>|</mo>}\mathfrak{w}{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left|\left(\mathfrak{w}-\frac{1}{q}\right){}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}]$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{w}{d}_q\mathfrak{w}\right)}^{1-\frac{1}{r}}{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathfrak{w}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}$

$+{\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{1}{q}-\mathfrak{w}\right)d_q\mathfrak{w}\right)}^{1-\frac{1}{r}}{\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{1}{q}-\mathfrak{w}\right){\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}]$

By applying (α, m)-convexity of ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r$ , we have

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{\left(\frac{1}{{\left[2\right]}_q^3}\right)}^{1-\frac{1}{r}}$

$\times [{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^{\alpha +1}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}+{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}m\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}-{\mathfrak{w}}^{\alpha +1}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}$

$(+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left(\frac{{\mathfrak{w}}^{\alpha }}{q}-{\mathfrak{w}}^{\alpha +1}\right){\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}$

${+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}m\left(\frac{1}{q}-\mathfrak{w}\right)\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w})}^{\frac{1}{r}}]$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q^{\frac{\mathrm{3<mo\; stretchy="false">(</mo>}r-\mathrm{1<mo\; stretchy="false">)</mo>}}{r}}}[{\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{r}}$

$+{\mathrm{<mo\; stretchy="false">(</mo>}{A}_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{A}_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{<mo\; stretchy="false">)</mo>}}^{\frac{1}{r}}]\mathrm{.}$

Thus, the proof is accomplished.

Remark 4 In Theorem 5, If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [14, Theorem 16].

Theorem 6  Under the assumption of Lemma 2, if r > 1 is a real number, if ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r$  is (a, m) convex function over $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then we have the following midpoint type inequality,

$r^{-1}+s^{-1}\mathrm{<mo>=</mo>1}$

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$\times [{\left(\frac{1}{{\left[2\right]}_q^{s+1}{\left[s+1\right]}_q}\right)}^{\frac{1}{s}}{\left(B_1\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{C}_1\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\right)}^{\frac{1}{r}}$

$+{\mathrm{<mo\; stretchy="false">(</mo>}\eta \mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}}^{\frac{1}{s}}{\left(B_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{C}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\right)}^{\frac{1}{r}}]\mathrm{,}\text{ (15)}$

where

$B_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^{\alpha }{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q}$

$B_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\mathfrak{w}}^{\alpha }{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[\alpha +1\right]}_q}-\frac{1}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q}$

$C_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[2\right]}_q}-\frac{1}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q}$

$C_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{q}{{\left[2\right]}_q}-\frac{1}{{\left[\alpha +1\right]}_q}+\frac{1}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q}$

$\eta \mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\left(\frac{1}{q}-\mathfrak{w}\right)}^s{d}_q\mathfrak{w}\mathrm{.}$

Proof. Taking absolute value of (6) and using the Hölder's inequality, we have

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathrm{<mo>|</mo>}\mathfrak{w}{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}{d}_q\mathfrak{w}$

$+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\left|\left(\mathfrak{w}-\frac{1}{q}\right){}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}]$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^s{d}_q\mathfrak{w}\right)}^{\frac{1}{s}}{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}$

$+{\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\left(\frac{1}{q}-\mathfrak{w}\right)}^s{d}_q\mathfrak{w}\right)}^{\frac{1}{s}}{\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}]$

By applying (α, m) convexity of ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r$ , we have

$\left|\mathfrak{F}\left(\frac{{\mathfrak{y}}_2+qm{\mathfrak{y}}_1}{{\left[2\right]}_q}\right)-\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}[{\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^s{d}_q\mathfrak{w}\right)}^{\frac{1}{s}}$

$\times {\left({\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}{\mathfrak{w}}^{\alpha }{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}+{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}m\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}$

$+{\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\left(\frac{1}{q}-\mathfrak{w}\right)}^s{d}_q\mathfrak{w}\right)}^{\frac{1}{s}}$

$\times {\left({\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}{\mathfrak{w}}^{\alpha }{\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}+{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}m\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}]$

$\mathrm{<mo>=</mo>}q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}$

$\times [{\left(\frac{1}{{\left[2\right]}_q^{s+1}{\left[s+1\right]}_q}\right)}^{\frac{1}{s}}{\left(B_1\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{C}_1\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\right)}^{\frac{1}{r}}$

$+{\mathrm{<mo\; stretchy="false">(</mo>}\eta \mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}}^{\frac{1}{s}}{\left(B_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r+m{C}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\right)}^{\frac{1}{r}}]\text{ (16)}$

Thus, the proof is accomplished.

Remark 5 In Theorem 6, we have

• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [14, Theorem 18].
• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ and later taking the limit as $q\to 1^{-}$ , then we find [42, Theorem 2.3].

Trapezoid type inequalities for (α, m)-convex functions

In this section, we prove Trapezoid type inequalities for differentiable (α, m) -convex functions.

Theorem 7  Under the assumption of Lemma 3, if ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo>|</mo>}$ is (a, m) convex function over $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then we have the following trapezoid type inequality:

$\left|-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}+\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}[\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|\mathrm{<mo\; stretchy="false">(</mo>}{K}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-K_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}$

$+m\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|\mathrm{<mo\; stretchy="false">(</mo>}{L}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-L_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}]\mathrm{,}\text{ (17)}$

where

$K_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{w}}^{\alpha }-{\left[2\right]}_q{\mathfrak{w}}^{\alpha +1}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{q^{\alpha +1}}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q{\left[\alpha +2\right]}_q}$

$K_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{w}}^{\alpha }-{\left[2\right]}_q{\mathfrak{w}}^{\alpha +1}\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{1}{{\left[\alpha +1\right]}_q}-\frac{{\left[2\right]}_q}{{\left[\alpha +2\right]}_q}-\frac{q^{\alpha +1}}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q{\left[\alpha +2\right]}_q}$

$L_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_0^{\frac{1}{{\left[2\right]}_q}}}\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\mathrm{<mo>=</mo>}\frac{q}{{\left[2\right]}_q^2}-\frac{q^{\alpha +1}}{{\left[2\right]}_q^{\alpha +1}{\left[\alpha +1\right]}_q{\left[\alpha +2\right]}_q}$

$L_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\displaystyle {\int }_{\frac{1}{{\left[2\right]}_q}}^1}\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}$

$\mathrm{<mo>=</mo>}\frac{q{\mathrm{<mo\; stretchy="false">[</mo>}\alpha \mathrm{<mo\; stretchy="false">]</mo>}}_q}{{\left[\alpha +1\right]}_q{\left[\alpha +2\right]}_q}-\frac{q}{{\left[2\right]}_q^2}+\frac{1}{{\left[\alpha +1\right]}_q{\left[2\right]}_q^{\alpha +1}}-\frac{1}{{\left[\alpha +2\right]}_q{\left[2\right]}_q^{\alpha +1}}$

Proof. By taking modulus in (11), and using (α, m) -convexity of ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo>|</mo>}$ , we have

$\left|-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}+\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}\left|{\displaystyle {\int }_0^1}{\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}\left(\begin{array}{c}{\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|{\mathfrak{w}}^{\alpha }\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}\\ +{\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|m\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}\end{array}\right)$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}[\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|\mathrm{<mo\; stretchy="false">(</mo>}{K}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-K_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+m\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|\mathrm{<mo\; stretchy="false">(</mo>}{L}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-L_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}]$

Thus, the proof is accomplished.

Remark 6 In Theorem 7, we have

• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [41, Theorem 4.1].
• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$  and later taking the limit as $q\to 1^{-}$ , then we find [43, Theorem 2.2].

Theorem 8  Under the assumption of Lemma 3, if ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r\mathrm{,}r\ge 1$ is (α, m) convex function over $\mathrm{<mo\; stretchy="false">[</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">]</mo>}$ , then we have the following trapezoid type inequality:

$\left|-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}+\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\left(\frac{2q}{{\left[2\right]}_q^2}\right)}^{1-\frac{1}{r}}$

$\times \left[{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r\mathrm{<mo\; stretchy="false">(</mo>}{K}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-K_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+m{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r\mathrm{<mo\; stretchy="false">(</mo>}{L}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-L_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\right]\mathrm{,}\text{ (18)}$

Proof. By taking modulus in (11) and using power mean inequality, we have

$\left|-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}+\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}\left|{\displaystyle {\int }_0^1}{\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{d}_q\mathfrak{w}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\left({\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}\right)}^{1-\frac{1}{r}}$

$\times {\left({\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{w}{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w}\right)}^{\frac{1}{r}}$

By applying (α, m)-convexity of ${\mathrm{<mo>|</mo>}}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}{\mathrm{<mo\; stretchy="false">)</mo><mo>|</mo>}}^r$ , we have

$\left|-\frac{\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}+q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}+\frac{1}{{\mathfrak{y}}_2-m{\mathfrak{y}}_1}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\mathfrak{F}{\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo>}}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\left({\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|d_q\mathfrak{w}\right)}^{1-\frac{1}{r}}$

$\times {\left(\begin{array}{c}{\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|{\mathfrak{w}}^{\alpha }{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w}\\ +{\displaystyle {\int }_0^1}\left|\mathrm{<mo\; stretchy="false">(</mo>1}-{\left[2\right]}_q\mathfrak{w}\mathrm{<mo\; stretchy="false">)</mo>}\right|m\mathrm{<mo\; stretchy="false">(</mo>1}-{\mathfrak{w}}^{\alpha }\mathrm{<mo\; stretchy="false">)</mo>}{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r{d}_q\mathfrak{w}\end{array}\right)}^{\frac{1}{r}}$

$\mathrm{<mo>=</mo>}\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q}{\left(\frac{2q}{{\left[2\right]}_q^2}\right)}^{1-\frac{1}{r}}$

$\times \left[{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r\mathrm{<mo\; stretchy="false">(</mo>}{K}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-K_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+m{\left|{}_{m{\mathfrak{y}}_1}{D}_q\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}\right|}^r\mathrm{<mo\; stretchy="false">(</mo>}{L}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}-L_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\right]$

Thus, the proof is accomplished.

Remark 7 In Theorem 8, we have

• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ , then we find [41, Theorem 4.2].
• If we set $\alpha \mathrm{<mo>=</mo>}m\mathrm{<mo>=</mo>1}$ and later taking the limit as $q\to 1^{-}$ , then we find [46, Theorem 6].

Application to special means

For any positive number ${\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\in ℝ$ , we consider the following means:

• The Arithmetic mean

$A\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{{\mathfrak{y}}_1+{\mathfrak{y}}_2}{2}$

• The Harmonic mean

$H\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{2{\mathfrak{y}}_1{\mathfrak{y}}_2}{{\mathfrak{y}}_1+{\mathfrak{y}}_2}$

• The Geometric mean

$G\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\sqrt{{\mathfrak{y}}_1{\mathfrak{y}}_2}\mathrm{.}$

Proposition 1 Let ${\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\in ℝ\mathrm{,}{\mathfrak{y}}_1\mathrm{<mo><</mo>}{\mathfrak{y}}_2\mathrm{,}\alpha \in \mathrm{0,1<mo\; stretchy="false">]</mo>,}m\in \mathrm{0,1<mo\; stretchy="false">]</mo>}$ and  $\mathrm{0<mo><</mo>}q\mathrm{<mo><</mo>1}$ . Then we have

$\left|H\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}-\frac{G^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}{\Upsilon }_1\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}$

$\times \frac{1}{{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+\frac{1}{m{\mathfrak{y}}_1^2}\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\text{ (19)}$

where ${\Upsilon }_1\mathrm{<mo>=</mo>}\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\frac{1}{\mathfrak{r}}{}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle {\sum }_{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{q^n}{q^n{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q^n\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1}\mathrm{.}$

Proof. The inequality (13) for function $\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{2q{\mathfrak{y}}_{2}m{\mathfrak{y}}_1}{{\left[2\right]}_q\mathfrak{r}}$ leads to required result. If we take ${\mathfrak{y}}_1\mathrm{<mo>=</mo>1,}{\mathfrak{y}}_2\mathrm{<mo>=</mo>2,}q\mathrm{<mo>=</mo>0.5,}m\mathrm{<mo>=</mo>0.5}$ and $\alpha \mathrm{<mo>=</mo>0.5}$ in (19), we get

$\left|H\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}-\frac{G^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}{\Upsilon }_1\right|\mathrm{<mo>=</mo>0.054}$

and

$\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}\left[\begin{array}{c}\frac{1}{{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+\frac{1}{m{\mathfrak{y}}_1^2}\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\end{array}\right]\mathrm{<mo>=</mo>0.071}$

Thus, $0.054\le \mathrm{0.071.}$

Proposition 2 Let ${\mathfrak{y}}_1\mathrm{,}{\mathfrak{y}}_2\in ℝ\mathrm{,}{\mathfrak{y}}_1\mathrm{<mo><</mo>}{\mathfrak{y}}_2\mathrm{,}\alpha \in \mathrm{0,1<mo\; stretchy="false">]</mo>,}m\in \mathrm{0,1<mo\; stretchy="false">]</mo>}$  and $\mathrm{0<mo><</mo>}q\mathrm{<mo><</mo>1}$ . Then we have

$\left|G^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}-\frac{A\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}{\Upsilon }_1\right|$

$\le \frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}A\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}$

$\times \frac{1}{{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+\frac{1}{m{\mathfrak{y}}_1^2}\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\text{ (20)}$

where ${\Upsilon }_1\mathrm{<mo>=</mo>}\frac{1}{\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{\displaystyle {\int }_{m{\mathfrak{y}}_1}^{{\mathfrak{y}}_2}}\frac{1}{\mathfrak{r}}{}_{m{\mathfrak{y}}_1}{d}_q\mathfrak{r}\mathrm{<mo>=</mo><mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}{\displaystyle {\sum }_{n\mathrm{<mo>=</mo>0}}^{\infty }}\frac{q^n}{q^n{\mathfrak{y}}_2+m\mathrm{<mo\; stretchy="false">(</mo>1}-q^n\mathrm{<mo\; stretchy="false">)</mo>}{\mathfrak{y}}_1}\mathrm{.}$

Proof. The inequality (13) for function $\mathfrak{F}\mathrm{<mo\; stretchy="false">(</mo>}\mathfrak{r}\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{q{\mathfrak{y}}_{2}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">(</mo>}qm{\mathfrak{y}}_1+{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">)</mo>}}{{\left[2\right]}_q\mathfrak{r}}$ leads to required result. If we take ${\mathfrak{y}}_1\mathrm{<mo>=</mo>1,}{\mathfrak{y}}_2\mathrm{<mo>=</mo>2,}q\mathrm{<mo>=</mo>0.5,}m\mathrm{<mo>=</mo>0.5}$ and $\alpha \mathrm{<mo>=</mo>0.5}$ in (20), we get

$\left|G^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}-\frac{A\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}{\Upsilon }_1\right|\mathrm{<mo>=</mo>0.188}$

and

$\frac{q\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2-m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}A\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}{G}^2\mathrm{<mo\; stretchy="false">(</mo>}{\mathfrak{y}}_2\mathrm{,}qm{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}{A\mathrm{<mo\; stretchy="false">(</mo>1,}q\mathrm{<mo\; stretchy="false">)</mo>}}\left[\begin{array}{c}\frac{1}{{\mathfrak{y}}_2\mathrm{<mo\; stretchy="false">(</mo>}q{\mathfrak{y}}_2+\mathrm{<mo\; stretchy="false">(</mo>1}-q\mathrm{<mo\; stretchy="false">)</mo>}m{\mathfrak{y}}_1\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo\; stretchy="false">(</mo>}{A}_1\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_3\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}+\frac{1}{m{\mathfrak{y}}_1^2}\mathrm{<mo\; stretchy="false">(</mo>}{A}_2\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo>}+A_4\mathrm{<mo\; stretchy="false">(</mo>}q\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}\end{array}\right]\mathrm{<mo>=</mo>3.27}$

Thus, $0.188\le \mathrm{3.27.}$

Conclusion

In the current study, we initially proved two quantum identities using the integration by parts method. Then, using these identities, we established some new Midpoint and Trapezoid type inequalities for differentiable (α, m) -convex functions, which was the main motivation of this paper. In upcoming directions, similar inequalities could be found for co-ordinated convex functions as well.