New Estimates of the q-Hermite–Hadamard Inequalities via Strong Convexity

Abstract

A refined version of the q-Hermite–Hadamard inequalities for strongly convex functions is introduced in this paper, utilizing both left and right q-integrals. Tighter bounds and more accurate estimates are derived by incorporating strong convexity. New q-trapezoidal and q-midpoint estimates are also presented to enhance the precision of the results. The improvements in the results compared to previous work are demonstrated through numerical examples in terms of precision and tighter bounds, and the advantages of using strongly convex functions are showcased.

1. Introduction

A function $\Psi :I\to \mathbb{R}$, defined on an interval $I\subseteq \mathbb{R}$, is convex if for all $\omega ,\zeta \in I$ and $\varphi \in [0,1]$, the inequality

$$\Psi (\varphi \omega +(1-\varphi \left)\zeta \right)\le \varphi \Psi \left(\omega \right)+(1-\varphi )\Psi \left(\zeta \right)$$

holds.

The concept of strong convexity, introduced by B.T. Polyak [1], extends convexity by incorporating a quadratic term. This refinement yields sharper results in various applications [2,3,4,5]. Specifically, $\Psi$ is said to be strongly convex with modulus $w>0$ if

$$\Psi (\varphi \omega +(1-\varphi )\zeta )\le \varphi \Psi \left(\omega \right)+(1-\varphi )\Psi \left(\zeta \right)-w\varphi (1-\varphi ){(\omega -\zeta )}^2$$

for all $\omega ,\zeta \in I$ and $\varphi \in [0,1]$.

While convex functions lie below the chord connecting any two points, strongly convex functions lie strictly below it by a parabolic gap determined by a parameter $w>0$. Hence, every strongly convex function is also convex. Since strong convexity is a stronger assumption than convexity, it allows for the derivation of sharper and more refined results. Moreover, strongly convex functions play an important role in optimization theory, as they guarantee the existence of a unique minimizer and enable faster convergence of many iterative methods. The improved bounds for q-integrals presented in this paper may therefore be useful in the analysis and design of q-analog algorithms in optimization, approximation theory, or numerical analysis.

The study of inequalities in mathematical analysis, particularly those involving convex functions, has drawn considerable attention. One of the most significant inequalities in this area is the Hermite–Hadamard inequality (H-H), first introduced by C. Hermite [6] and J. Hadamard [7], which provides bounds on the mean value of a convex function over a given interval. This inequality is a notable extension of Jensen’s inequality [8], and it has found wide applications in the study of convex functions. In 2018, N. Alp et al. [9] provided a generalization of the classical H-H inequality by introducing its q-analogue for convex functions via the use of left-sided q-integrals. More precisely, for any convex and differentiable function $\Psi :[\varrho ,\kappa ]\to \mathbb{R}$ where $0<q<1$, the inequality below is satisfied:

$$\Psi \left({\displaystyle \frac{q\varrho +\kappa }{{\left[2\right]}_q}}\right)\le {\displaystyle \frac{1}{\kappa -\varrho }}{\int }_{\varrho }^{\kappa }\Psi \left(\omega \right){}_{\varrho }d_q\omega \le {\displaystyle \frac{q\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{{\left[2\right]}_q}},$$

(1)

where ${\left[2\right]}_q=1+q$. This formulation adds a new dimension to the study of convexity by incorporating q-integrals into the analysis.

More recently, S. Bermudo et al. [10] applied right q-integrals to derive an alternative form of the q-H-H inequality:

$$\Psi \left({\displaystyle \frac{\varrho +q\kappa }{{\left[2\right]}_q}}\right)\le {\displaystyle \frac{1}{\kappa -\varrho }}{\int }_{\varrho }^{\kappa }\Psi \left(\omega \right){}^{\kappa }d_q\omega \le {\displaystyle \frac{\Psi \left(\varrho \right)+q\Psi \left(\kappa \right)}{{\left[2\right]}_q}}.$$

(2)

By combining (1) and (2), Bermudo et al. [10] derived a symmetric form of the q-H-H inequality:

$$\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)\le {\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\omega \right){}_{\varrho }d_q\omega +{\int }_{\varrho }^{\kappa }\Psi \left(\omega \right){}^{\kappa }d_q\omega \right)\le {\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}.$$

(3)

Details regarding the left- and right-sided bounds of (1)–(3), as well as generalizations involving convexity concepts within the framework of q-calculus, can be found in [11,12,13,14,15,16,17,18,19,20,21,22].

The H-H inequality has emerged as a key subject of study due to its versatility in extending convexity frameworks and producing sharper estimates. Recent advancements incorporating q-calculus, particularly through the use of q-integrals, have yielded significant developments in integral inequalities. For further details, see [23,24,25]. Currently, q-H-H inequalities for strongly convex functions have also been investigated (see [26,27]), further enriching the analytical framework and opening avenues for broader applications in optimization and mathematical modeling.

Calculus, a cornerstone of modern mathematics, is primarily concerned with the study of functions and their rates of change. The foundational theories were independently established in the 17th century by Isaac Newton and Gottfried Wilhelm Leibniz, forming the basis for classical calculus. Building upon these foundations, Leonhard Euler (1707–1783) pioneered early ideas related to what is now known as quantum calculus (q-calculus), which extends traditional calculus by eliminating the dependence on limits. In the early 20th century, Jackson [28,29] significantly advanced this theory by developing a formal structure for q-calculus.

Further progress was made in the early 21st century, notably through the comprehensive work of Kac and Cheung [30], who provided a unified and accessible account of the fundamental principles of q-calculus. Additional discussions and developments can be found in [31,32], which explore various applications and properties of the theory. In later contributions, particularly those of Tariboon and Ntouyas [33,34], the framework of q-calculus was extended to accommodate continuous functions on closed intervals. This extension further enriched the analytical tools available for studying discrete and quantum models.

This study investigates strongly convex functions as a refinement of convex functions within the framework of Hermite–Hadamard-type inequalities under q-calculus. Numerical results reveal that strong convexity leads to tighter bounds and more precise estimates. Section 5 presents a comparison with previous work, highlighting the improved sharpness of the proposed inequalities. Graphical illustrations further demonstrate the advantages of adopting strong convexity in the q-calculus setting.

The organization of this paper is as follows. Section 2 introduces the fundamental concepts of q-calculus. Section 3 is devoted to deriving q-trapezoidal and q-midpoint-type estimates for inequality (3), along with a discussion of their connections to existing results. Section 5 provides numerical examples that demonstrate the validity and improved accuracy of the proposed inequalities. Finally, Section 6 summarizes the main contributions and outlines possible directions for future research.

2. Preliminaries

This section provides key definitions and foundational properties of q-calculus that are central to our analysis. Throughout this discussion, we consider $\varrho <\kappa$ and $0<q<1$.

The quantity known as the q-number of $\gamma$ is given by:

$$\begin{array}{c}\hfill {\left[\gamma \right]}_q:={\displaystyle \frac{1-q^{\gamma }}{1-q}},\mathrm{if}\gamma \in {\mathbb{R}}^{+}.\end{array}$$

Note that if $\gamma \in \mathbb{N}$, then ${\left[\gamma \right]}_q=1+q+\cdots +q^{\gamma -2}+q^{\gamma -1}.$

Definition 1 

([33]). Let $\Psi :[\varrho ,\kappa ]\to \mathbb{R}$ be a continuous function. The $q_{\varrho }$-derivative of $\Psi$ at $\omega \in [\varrho ,\kappa ]$ is defined as

$${}_{\varrho }D_q\Psi \left(\omega \right)={\displaystyle \frac{\Psi \left(\omega \right)-\Psi (q\omega +(1-q\left)\varrho \right)}{(1-q)(\omega -\varrho )}},\omega \ne \varrho .$$

(4)

If $\omega =\varrho$, we define

$${}_{\varrho }D_q\Psi \left(\varrho \right)=\underset{\omega \to \varrho }{lim}\varrho D_q\Psi \left(\omega \right),$$

if the limit exists and is finite. For $\varrho =0$, this reduces to the q-Jackson derivative:

$$D_q\Psi \left(\omega \right)={\displaystyle \frac{\Psi \left(\omega \right)-\Psi \left(q\omega \right)}{(1-q)\omega }},\omega \ne 0,$$

as discussed in [28,29,30,33,34].

Definition 2 

([10]). For a continuous function $\Psi :[\varrho ,\kappa ]\to \mathbb{R}$, the $q^{\kappa }$-derivative at $\omega \in [\varrho ,\kappa ]$ is given by

$${}^{\kappa }D_q\Psi \left(\omega \right)={\displaystyle \frac{\Psi (q\omega +(1-q\left)\kappa \right)-\Psi \left(\omega \right)}{(1-q)(\kappa -\omega )}},\omega \ne \kappa .$$

(5)

If $\omega =\kappa$, we define

$${}^{\kappa }D_q\Psi \left(\kappa \right)=\underset{\omega \to \kappa }{lim}{}^{\kappa }D_q\Psi \left(\omega \right),$$

if the limit exists and is finite.

Definition 3 

[33]). For a continuous function $\Psi :[\rho ,\kappa ]\to \mathbb{R}$, the $q_{\rho }$-integral of $\Psi$ over $[\rho ,\omega ]$ is defined as

$${\int }_{\rho }^{\omega }\Psi \left(\varphi \right){}_{\rho }d_q\varphi =(1-q)(\omega -\rho ){\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi \left(q^n\omega +(1-q^n)\rho \right).$$

(6)

For $\rho =0$, the integral reduces to the following form:

$${\int }_0^{\omega }\Psi \left(\varphi \right){}_{0}d_q\varphi =(1-q)\omega {\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi \left(q^n\omega \right),$$

which is the $q_{\rho }$-integral, as discussed in [28,29,30,33,34].

Definition 4 

([10]). For a continuous function $\Psi :[\rho ,\kappa ]\to \mathbb{R}$, the $q^{\kappa }$-integral over $[\omega ,\kappa ]$ is given by

$${\int }_{\omega }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi =(1-q)(\kappa -\omega ){\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi \left(q^n\omega +(1-q^n)\kappa \right).$$

(7)

When $\omega =0$, this simplifies to

$${\int }_0^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi =(1-q)\kappa {\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi \left((1-q^n)\kappa \right),$$

which is the $q^{\kappa }$-integral.

Example 1. 

Let $s\in {\mathbb{R}}^{+}$. Consider the function $\mathfrak{F}\left(\tau \right)={\tau }^s$ for $\tau \in [0,1]$. Then the ${}_{0}d_q$-integral of $\mathfrak{F}$ satisfies

$$\begin{array}{cc}\hfill {\int }_0^1{\tau }^s{}_{0}d_q\tau & =(1-q)\left(1-0\right){\displaystyle {\displaystyle \sum _{n=0}^{\infty }}}{q}^n\mathfrak{F}\left(q^n+\left(1-q^n\right)0\right)\hfill \\ \hfill & =(1-q){\displaystyle \sum _{n=0}^{\infty }}{q}^n\left(q^{sn}\right)\hfill \\ \hfill & =(1-q){\displaystyle \sum _{n=0}^{\infty }}{q}^{(s+1)n}\hfill \\ \hfill & ={\displaystyle \frac{1}{{[s+1]}_q}}.\hfill \end{array}$$

Lemma 1 

([35]). [q-Hölder inequality] Suppose that f and w are q-integrable on $[{\alpha }_1,{\alpha }_2]$, and that ${|f|}^{\mu }$ and ${|g|}^{\nu }$ are q-integrable on $[{\alpha }_1,{\alpha }_2]$. If $\nu >1$ and $1/\mu +1/\nu =1$, then

$${\int }_{{\alpha }_1}^{{\alpha }_2}|f\left(t\right)g\left(t\right)| {}_{{\alpha }_1}d_{q}t\le {\left({\int }_{{\alpha }_1}^{{\alpha }_2}{|f\left(t\right)|}^{\mu } {}_{{\alpha }_1}d_{q}t\right)}^{{\displaystyle \frac{1}{\mu }}}{\left({\int }_{{\alpha }_1}^{{\alpha }_2}{|g\left(t\right)|}^{\nu } {}_{{\alpha }_1}d_{q}t\right)}^{{\displaystyle \frac{1}{\nu }}}.$$

3. q-Trapezoidal Inequalities

We begin this section by establishing Lemma 2 to derive the right-hand bounds of inequality (3) using differentiable strongly convex functions.

Lemma 2. 

Let $\Psi :[\varrho ,\kappa ]\to \mathbb{R}$ be a q-differentiable function. If both the left q-derivative ${}_{\varrho }D_q\Psi$ and the right q-derivative ${}^{\kappa }D_q\Psi$ are continuous and integrable over $[\varrho ,\kappa ]$, then

$$\begin{array}{cc}\hfill & {\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left({}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho ){-}^{\kappa }{D}_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right)d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi \right).\hfill \end{array}$$

(8)

Proof. 

By Definition 1, we have

$$\begin{array}{cc}\hfill I_1& :={\int }_0^{1}q\varphi \left({}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right)d_q\varphi \hfill \\ \hfill & ={\int }_0^{1}q\varphi {\displaystyle \frac{\Psi (\varphi \kappa +(1-\varphi \left)\varrho \right)-\Psi (q\varphi \kappa +(1-q\varphi \left)\varrho \right)}{(1-q)\varphi (\kappa -\varrho )}}{d}_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{q}{(\kappa -\varrho )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^n\kappa +(1-q^n)\varrho )-{\displaystyle \frac{q}{(\kappa -\varrho )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^{n+1}\kappa +(1-q^{n+1})\varrho )\hfill \\ \hfill & ={\displaystyle \frac{1}{(\kappa -\varrho )}}\Psi \left(\kappa \right)-{\displaystyle \frac{1-q}{(\kappa -\varrho )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^n\kappa +(1-q^n)\varrho )\hfill \\ \hfill & ={\displaystyle \frac{1}{(\kappa -\varrho )}}\Psi \left(\kappa \right)-{\displaystyle \frac{1}{{(\kappa -\varrho )}^2}}{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi .\hfill \end{array}$$

Similarly, by Definition 2, we have

$$\begin{array}{cc}\hfill I_2& :={\int }_0^{1}q\varphi \left({}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right)d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{1}{{(\kappa -\varrho )}^2}}{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi -{\displaystyle \frac{1}{(\kappa -\varrho )}}\Psi \left(\varrho \right).\hfill \end{array}$$

Then it follows that

$$\begin{array}{cc}\hfill {\displaystyle \frac{(\kappa -\varrho )}{2}}(I_1-I_2)& ={\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi \right),\hfill \end{array}$$

and this completes the proof. □

The following theorem establishes the right-hand bound of inequality (3) by employing Lemma 2 and the property of strong convexity.

Theorem 1. 

Under the assumptions of Lemma 2, the following inequalities are satisfied if ${|}_{\varrho }{D}_q\Psi |$ and $|{}^{\kappa }D_q\Psi |$ are strongly convex with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left(|{}_{\varrho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\varrho \right)|\right)+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\left(|{}_{\varrho }D_q\Psi \left(\varrho \right)|+|{}^{\kappa }D_q\Psi \left(\kappa \right)|\right)\right)\hfill \\ \hfill & -\left({\displaystyle \frac{q^{4}w{(\kappa -\varrho )}^3}{{\left[3\right]}_q{\left[4\right]}_q}}\right).\hfill \end{array}$$

(9)

Proof. 

Applying Lemma 2, together with the strong convexity of $|{}_{\varrho }D_q\Psi |$ and ${|}^{\kappa }{D}_q\Psi |$, yields that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left(\left|{}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right|+\left|{}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right|\right)d_q\varphi \hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left(\varphi |{}_{\varrho }D_q\Psi \left(\kappa \right)|+(1-\varphi )|{}_{\varrho }D_q{\Psi \left(\varrho \right)|-w\varphi (1-\varphi )(\kappa -\varrho )}^2\right)d_q\varphi \hfill \\ \hfill & +{\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left(\varphi |{}^{\kappa }D_q\Psi \left(\varrho \right)|+(1-\varphi )|{}^{\kappa }D_q{\Psi \left(\kappa \right)|-w\varphi (1-\varphi )(\kappa -\varrho )}^2\right)d_q\varphi \hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^1\left(q{\varphi }^2|{}_{\varrho }D_q\Psi \left(\kappa \right)|+(q\varphi -q{\varphi }^2)|{}_{\varrho }D_q\Psi \left(\varrho \right)|-wq({\varphi }^2-{\varphi }^3){(\kappa -\varrho )}^2\right)d_q\varphi \hfill \\ \hfill & +{\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^1\left(q{\varphi }^2|{}^{\kappa }D_q\Psi \left(\varrho \right)|+(q\varphi -q{\varphi }^2)|{}^{\kappa }D_q\Psi \left(\kappa \right)|-wq({\varphi }^2-{\varphi }^3){(\kappa -\varrho )}^2\right)d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{(\kappa -\varrho )}{2}}\left({\displaystyle \frac{q|{}_{\varrho }D_q\Psi \left(\kappa \right)|}{{\left[3\right]}_q}}+\left({\displaystyle \frac{q}{{\left[2\right]}_q}}-{\displaystyle \frac{q}{{\left[3\right]}_q}}\right)|{}_{\varrho }D_q\Psi \left(\varrho \right)|-wq\left({\displaystyle \frac{1}{{\left[3\right]}_q}}-{\displaystyle \frac{1}{{\left[4\right]}_q}}\right){(\kappa -\varrho )}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{(\kappa -\varrho )}{2}}\left({\displaystyle \frac{q|{}^{\kappa }D_q\Psi \left(\varrho \right)|}{{\left[3\right]}_q}}+\left({\displaystyle \frac{q}{{\left[2\right]}_q}}-{\displaystyle \frac{q}{{\left[3\right]}_q}}\right)|{}^{\kappa }D_q\Psi \left(\kappa \right)|-wq\left({\displaystyle \frac{1}{{\left[3\right]}_q}}-{\displaystyle \frac{1}{{\left[4\right]}_q}}\right){(\kappa -\varrho )}^2\right)\hfill \\ \hfill & ={\displaystyle \frac{(\kappa -\varrho )}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left(|{}_{\varrho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\varrho \right)|\right)+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\left(|{}_{\varrho }D_q\Psi \left(\varrho \right)|+|{}^{\kappa }D_q\Psi \left(\kappa \right)|\right)\right)\hfill \\ \hfill & -\left({\displaystyle \frac{q^{4}w{(\kappa -\varrho )}^3}{{\left[3\right]}_q{\left[4\right]}_q}}\right).\hfill \end{array}$$

This concludes the proof. □

The next theorem provides the right-hand bound of inequality (3) using Lemma 2, Theorem 1, and the strong convexity property.

Theorem 2. 

Under the assumptions of Lemma 2, the following inequalities are satisfied if ${|}_{\varrho }{D}_q{\Psi |}^{\mu }$ and ${|}^{\kappa }{D}_q{\Psi |}^{\mu }$, $\mu \ge 1$, are strongly convex functions with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{q(\kappa -\varrho )}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+q^2{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+q^2{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}).\hfill \end{array}$$

(10)

Proof. 

It follows from Lemmas 1 and 2, and the strong convexity of $|{}_{\varrho }D_q{\Psi |}^{\mu }$ and $|{}^{\kappa }D_q{\Psi |}^{\mu }$ that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left(\left|{}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right|+\left|{}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right|\right)d_q\varphi \hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^{1}q\varphi d_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\int }_0^{1}q\varphi {\left|{\phantom{­}}_{\varrho }{D}_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right|}^{\mu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^{1}q\varphi d_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\int }_0^{1}q\varphi {\left|{\phantom{­}}^{\kappa }{D}_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right|}^{\mu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^{1}q\varphi d_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\int }_0^{1}q\varphi (\varphi |{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+(1-\varphi ){|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }-w \varphi (1-\varphi )\hfill \\ \hfill & {(\kappa -\varrho )}^2)d_q\varphi {)}^{{\displaystyle \frac{1}{\mu }}}+{\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^{1}q\varphi d_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\int }_0^{1}q\varphi (\varphi |{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+(1-\varphi ){|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & -w \varphi (1-\varphi ){(\kappa -\varrho )}^2)d_q\varphi {)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & ={\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\displaystyle \frac{q}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{q|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }}{{\left[3\right]}_q}}+\left({\displaystyle \frac{q}{{\left[2\right]}_q}}-{\displaystyle \frac{q}{{\left[3\right]}_q}}\right){|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }-wq\left({\displaystyle \frac{1}{{\left[3\right]}_q}}-{\displaystyle \frac{1}{{\left[4\right]}_q}}\right)\hfill \\ \hfill & {(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\displaystyle \frac{q}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{q|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }}{{\left[3\right]}_q}}+\left({\displaystyle \frac{q}{{\left[2\right]}_q}}-{\displaystyle \frac{q}{{\left[3\right]}_q}}\right){|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & -wq\left({\displaystyle \frac{1}{{\left[3\right]}_q}}-{\displaystyle \frac{1}{{\left[4\right]}_q}}\right){(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & ={\displaystyle \frac{q(\kappa -\varrho )}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+q^2{|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+q^2{|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}).\hfill \end{array}$$

This concludes the proof. □

The following theorem establishes the right-hand bound of inequality (3) by applying Lemma 2, Theorem 1, and the strong convexity property.

Theorem 3. 

Under the assumptions of Lemma 2, the following inequalities are satisfied if ${|}_{\varrho }{D}_q{\Psi |}^{\mu }$ and $|{}^{\kappa }D_q{\Psi |}^{\mu }$, $\mu >1$ are strongly convex functions on $[\varrho ,\kappa ]$ with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{q(\kappa -\varrho )}{2{\left({[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\{{\left({\displaystyle \frac{|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+q{|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+q{|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\},\hfill \end{array}$$

(11)

where $1/\mu +1/\nu =1$.

Proof. 

By applying Lemmas 1 and 2, and using the fact that ${|}_{\varrho }{D}_q{\Psi |}^{\mu }$ and ${|}^{\kappa }{D}_q{\Psi |}^{\mu }$ are strongly convex functions, we deduce that

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{\Psi \left(\varrho \right)+\Psi \left(\kappa \right)}{2}}-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\int }_0^{1}q\varphi \left(\left|{}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right|+\left|{}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right|\right)d_q\varphi \hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^1{\left(q\varphi \right)}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\int }_0^1{\left|{\phantom{­}}_{\varrho }{D}_q\Psi (\varphi \kappa +(1-\varphi )\varrho )\right|}^{\mu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^1{\left(q\varphi \right)}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\int }_0^1{\left|{\phantom{­}}^{\kappa }{D}_q\Psi (\varphi \varrho +(1-\varphi )\kappa )\right|}^{\mu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & \le {\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^1{\left(q\varphi \right)}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\int }_0^1(\varphi |{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+(1-\varphi ){|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & -w \varphi (1-\varphi ){(\kappa -\varrho )}^2)d_q\varphi {)}^{{\displaystyle \frac{1}{\mu }}}+{\displaystyle \frac{(\kappa -\varrho )}{2}}{\left({\int }_0^1{\left(q\varphi \right)}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\int }_0^1(\varphi |{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & +(1-\varphi )|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right){|}^{\mu }-w \varphi (1-\varphi ){(\kappa -\varrho )}^2)d_q\varphi {)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & ={\displaystyle \frac{q(\kappa -\varrho )}{2}}{\left({\displaystyle \frac{1}{{[\nu +1]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }}{{\left[2\right]}_q}}+\left(1-{\displaystyle \frac{1}{{\left[2\right]}_q}}\right){|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }-w\left({\displaystyle \frac{1}{{\left[2\right]}_q}}-{\displaystyle \frac{1}{{\left[3\right]}_q}}\right)\hfill \\ \hfill & {(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\displaystyle \frac{q(\kappa -\varrho )}{2}}{\left({\displaystyle \frac{1}{{[\nu +1]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }}{{\left[2\right]}_q}}+\left(1-{\displaystyle \frac{1}{{\left[2\right]}_q}}\right){|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & -w\left({\displaystyle \frac{1}{{\left[2\right]}_q}}-{\displaystyle \frac{1}{{\left[3\right]}_q}}\right){(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & ={\displaystyle \frac{q(\kappa -\varrho )}{2{\left({[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\{{\left({\displaystyle \frac{|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+q{|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+q{|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\}.\hfill \end{array}$$

This concludes the proof. □

4. q-Midpoint Inequalities

This section derives enhanced left-hand bounds for inequality (3) using differentiable strongly convex functions. Lemma 3 is established below to support the proof.

Lemma 3. 

Let $\Psi :[\varrho ,\kappa ]\to \mathbb{R}$ be a q-differentiable function. If both the left q-derivative ${}_{\varrho }D_q\Psi$ and the right q-derivative ${}^{\kappa }D_q\Psi$ are continuous and integrable over $[\varrho ,\kappa ]$, then:

$$\begin{array}{cc}\hfill & \Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\hfill \\ \hfill & ={\displaystyle \frac{(\kappa -\varrho )}{2}}\{{\int }_0^{{\displaystyle \frac{1}{2}}}q\varphi {}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )d_q\varphi +{\int }_{{\displaystyle \frac{1}{2}}}^1(q\varphi -1){}_{\varrho }D_q\Psi (\varphi \kappa +(1-\varphi )\varrho )d_q\varphi \hfill \\ \hfill & +{\int }_0^{{\displaystyle \frac{1}{2}}}(-q\varphi ){}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )d_q\varphi +{\int }_{{\displaystyle \frac{1}{2}}}^1(1-q\varphi ){}^{\kappa }D_q\Psi (\varphi \varrho +(1-\varphi )\kappa )d_q\varphi \}.\hfill \end{array}$$

(12)

Proof. 

By Definition 1, we have

$$\begin{array}{cc}\hfill I_1& :={\int }_0^{{\displaystyle \frac{1}{2}}}q\varphi {}_{\kappa }D_q\Psi (\varphi \rho +(1-\varphi )\kappa )d_q\varphi +{\int }_{{\displaystyle \frac{1}{2}}}^1(q\varphi -1){}_{\kappa }D_q\Psi (\varphi \rho +(1-\varphi )\kappa )d_q\varphi \hfill \\ \hfill & ={\int }_0^{1}q\varphi {}_{\kappa }D_q\Psi (\varphi \rho +(1-\varphi )\kappa )d_q\varphi -{\int }_{{\displaystyle \frac{1}{2}}}^1{}_{\kappa }D_q\Psi (\varphi \rho +(1-\varphi )\kappa )d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{q}{(\rho -\kappa )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^n\rho +(1-q^n)\kappa )-{\displaystyle \frac{1}{(\rho -\kappa )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^n\rho +(1-q^n)\kappa )\hfill \\ \hfill & +{\displaystyle \frac{1}{(\rho -\kappa )}}\Psi \left(\rho \right)-{\displaystyle \frac{(1-q)(\rho -\frac{\kappa +\rho }{2})}{(\rho -\frac{\kappa +\rho }{2})}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n{}_{\kappa }D_q\Psi \left(q^n\rho +(1-q^n){\displaystyle \frac{\kappa +\rho }{2}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{(\rho -\kappa )}}\Psi \left(\rho \right)-{\displaystyle \frac{1-q}{(\rho -\kappa )}}{\displaystyle \sum _{n=0}^{\infty }}{q}^n\Psi (q^n\rho +(1-q^n)\kappa )-{\displaystyle \frac{1}{\rho -\kappa }}{\int }_{{\displaystyle \frac{\kappa +\rho }{2}}}^{\rho }{}_{\kappa }D_q\Psi \left(\varphi \right){}_{\kappa }d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{1}{(\rho -\kappa )}}\Psi \left({\displaystyle \frac{\kappa +\rho }{2}}\right)-{\displaystyle \frac{1}{{(\rho -\kappa )}^2}}{\int }_{\kappa }^{\rho }\Psi \left(\varphi \right){}_{\kappa }d_q\varphi .\hfill \end{array}$$

Similarly, Definition 2, we have

$$\begin{array}{cc}\hfill I_2& :={\int }_0^{{\displaystyle \frac{1}{2}}}(-q\varphi ){}^{\rho }D_q\Psi (\varphi \kappa +(1-\varphi )\rho )d_q\varphi +{\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\rho }{D}_q\Psi (\varphi \kappa +(1-\varphi )\rho )d_q\varphi \hfill \\ \hfill & ={\displaystyle \frac{1}{\rho -\kappa }}\Psi \left({\displaystyle \frac{\kappa +\rho }{2}}\right)-{\displaystyle \frac{1}{{(\rho -\kappa )}^2}}{\int }_{\kappa }^{\rho }\Psi \left(\varphi \right){}^{\rho }d_q\varphi .\hfill \end{array}$$

Then it follows that

$${\displaystyle \frac{(\rho -\kappa )}{2}}(I_1+I_2)=\Psi \left({\displaystyle \frac{\kappa +\rho }{2}}\right)-{\displaystyle \frac{1}{2(\rho -\kappa )}}\left({\int }_{\kappa }^{\rho }\Psi {\left(\varphi \right)}_{\kappa }{d}_q\varphi +{\int }_{\kappa }^{\rho }\Psi \left(\varphi \right){}^{\rho }d_q\varphi \right),$$

and the proof is completed. □

The next theorem establishes the left-hand bound of inequality (3) via Lemma 3 and strong convexity.

Theorem 4. 

Under the assumptions of Lemma 3, the following inequalities are satisfied if ${|}_{\varrho }{D}_q\Psi |$ and ${|}^{\kappa }{D}_q\Psi |$ are strongly convex functions with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{\kappa -\varrho }{2}}\{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q}{{\left[3\right]}_q}}\right)(|{}_{\varrho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\varrho \right)|)+\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\right)(|{}_{\varrho }D_q\Psi \left(\varrho \right)|\hfill \\ \hfill & +|{}^{\kappa }D_q\Psi \left(\kappa \right)|)-w\left({\displaystyle \frac{q+2q^2+{\left[4\right]}_q}{4{\left[2\right]}_q{\left[3\right]}_q}}+{\displaystyle \frac{q^4}{{\left[3\right]}_q{\left[4\right]}_q}}\right){(\kappa -\varrho )}^2\}.\hfill \end{array}$$

(13)

Proof. 

The proof follows easily by applying the procedure used in Theorem 1. □

The next theorem provides the left-hand bound of inequality (3) via Lemma 3, Theorem 1, and strong convexity.

Theorem 5. 

Under the assumptions of Lemma 3, the following inequalities are satisfied if $|\varrho D_q{\Psi |}^{\mu }$ and $|{}^{\kappa }D_q{\Psi |}^{\mu }$, $\mu \ge 1$ with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\left\{{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}\right({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}\hfill \\ \hfill & |{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }-w{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{16{\left[2\right]}_q{\left[3\right]}_q{\left[4\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}\hfill \\ \hfill & |{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-w{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{16{\left[2\right]}_q{\left[3\right]}_q{\left[4\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\}.\hfill \end{array}$$

(14)

Proof. 

The proof follows easily by applying the procedure used in Theorem 2. □

The following theorem gives the left-hand bound of inequality (3) via Lemma 3, Theorem 1, and strong convexity.

Theorem 6. 

Under the assumptions of Lemma 3, the following inequalities are satisfied if ${|}_{\varrho }{D}_q{\Psi |}^{\mu }$ and $|{}^{\kappa }D_q{\Psi |}^{\mu }$, $\mu >1$, are strongly convex functions on $[\varrho ,\kappa ]$ with respect to $w>0$:

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}_{\varrho }d_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ \hfill & \le \left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\left\{\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right)\right({\displaystyle \frac{1}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & +{\displaystyle \frac{q}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\varrho \right)|}^{\mu }-{\displaystyle \frac{{\left[2\right]}_{q}qw}{8{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right)({\displaystyle \frac{1}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & +{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}\hfill \\ \hfill & {\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-{\displaystyle \frac{w{\left[2\right]}_{q}q}{8{\left[3\right]}_q}}{(\kappa -\varrho )}^2\right)}^{{\displaystyle \frac{1}{\mu }}}\}.\hfill \end{array}$$

(15)

where $1/\mu +1/\nu =1$.

Proof. 

The proof follows easily by applying the procedure used in Theorem 3. □

Remark 1. 

Similarly to the proofs of Theorems 1–6, we recover the classical structure used for convex functions in Chasreechai et al. [21] by setting $w=0$ in each of our results. All of our results coincide with those of Chasreechai et al., except for their Theorem 4 and our Theorem 4, which are different. Indeed, by setting $w=0$ in Theorem 4, we obtain the inequality

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ & \le {\displaystyle \frac{\kappa -\varrho }{2}}\{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q}{{\left[3\right]}_q}}\right)(|{}_{\varrho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\varrho \right)|)\hfill \\ & +\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\right)(|{}_{\varrho }D_q\Psi \left(\varrho \right)|+|{}^{\kappa }D_q\Psi \left(\kappa \right)|)\}.\hfill \end{array}$$

(16)

On the other hand, Theorem 4 of [21] asserts

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi {\left(\varphi \right)}_{\varrho }{d}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){}^{\kappa }d_q\varphi \right)\right|\hfill \\ & \le {\displaystyle \frac{\kappa -\varrho }{2}}\{\left({\displaystyle \frac{3}{4\left({\left[4\right]}_q+q{\left[2\right]}_q\right)}}\right)\left(|{}_{\varrho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\varrho \right)|\right)\hfill \\ & +\left({\displaystyle \frac{5q^2+4q-2q^3-1}{4\left({\left[4\right]}_q+q{\left[2\right]}_q\right)}}\right)|{}_{\varrho }D_q\Psi \left(\varrho \right)|+\left({\displaystyle \frac{5q^2+4q-2q^3-1}{8\left({\left[4\right]}_q+q{\left[2\right]}_q\right)}}\right)|{}^{\kappa }D_q\Psi \left(\kappa \right)|\}.\hfill \end{array}$$

(17)

Moreover, we provide Example 2 to show that the inequality in their Theorem 4 does not generally hold.

5. Numerical Examples

In this section, we present numerical examples to verify the validity of our newly established inequalities. These examples are constructed to examine whether the proposed results align with theoretical expectations and to compare them with those obtained in previous studies. Graphs based on these examples are generated using MATLAB Online (https://matlab.mathworks.com) to visually support the analytical findings and highlight the advantages of the proposed approach.

Example 2. 

Let $\Psi :[0,1]\to \mathbb{R}$ defined by $\Psi (\Delta )=\Delta$ with $q=1/2$. Then ${|}_{\varrho }{D}_q\Psi (\Delta )|=1$ and ${|}^{\kappa }{D}_q\Psi (\Delta )|=1$ are convex functions. Then the left-hand side of the inequality (16) becomes

$$\begin{array}{cc}\hfill & \left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\kappa -\varrho )}}\left({\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){\phantom{­}}_{\varrho }{D}_q\varphi +{\int }_{\varrho }^{\kappa }\Psi \left(\varphi \right){\phantom{­}}^{\kappa }{D}_q\varphi \right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{2}}\left({\int }_0^1\varphi {\phantom{­}}_0{d}_{{\displaystyle \frac{1}{2}}}\varphi +{\int }_0^1\varphi {\phantom{­}}^1{d}_{{\displaystyle \frac{1}{2}}}\varphi \right)\right|\hfill \\ \hfill & \approx 0.6667,\hfill \end{array}$$

(18)

and the right-hand side becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\kappa -\rho }{2}}\{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q}{{\left[3\right]}_q}}\right)(|{}_{\rho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\rho \right)|)\hfill \\ \hfill & +\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\right)(|{}_{\rho }D_q\Psi \left(\rho \right)|+|{}^{\kappa }D_q\Psi \left(\kappa \right)|)\}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}+{\displaystyle \frac{\left(\frac{1}{2}\right)}{{\left[3\right]}_{\frac{1}{2}}}}+{\displaystyle \frac{{\left(\frac{1}{2}\right)}^3}{{\left[2\right]}_{\frac{1}{2}}{\left[3\right]}_{\frac{1}{2}}}}\approx 0.8333,\hfill \end{array}$$

(19)

and the right-hand side expression in Theorem 4 of [21] simplifies to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\kappa -\rho }{2}}\{\left({\displaystyle \frac{3}{4({\left[4\right]}_q+q{\left[2\right]}_q)}}\right)(|{}_{\rho }D_q\Psi \left(\kappa \right)|+|{}^{\kappa }D_q\Psi \left(\rho \right)|)\hfill \\ & +\left({\displaystyle \frac{5q^2+4q-2q^3-1}{4({\left[4\right]}_q+q{\left[2\right]}_q)}}\right)|{}_{\rho }D_q\Psi \left(\rho \right)|+\left({\displaystyle \frac{5q^2+4q-2q^3-1}{8({\left[4\right]}_q+q{\left[2\right]}_q)}}\right)|{}^{\kappa }D_q\Psi \left(\kappa \right)|\}\hfill \\ & \approx 0.4286.\hfill \end{array}$$

(20)

From inequality (16), together with (18) and (19), it follows that $0.6667<0.8333$, and thus the inequality is satisfied. In contrast, when employing Theorem 4 of [21], along with (18) and (20), we obtain $0.6667>0.4286$, which indicates that the inequality is not fulfilled.

Example 3. 

Let $\Psi :[0,1]\to \mathbb{R}$ defined by $\Psi (\Delta )={\Delta }^3/3+{\Delta }^2/2$. Then ${|}_{\varrho }{D}_q\Psi (\Delta )|=1/3\left({\Delta }^2(1+q+q^2)+\varrho \Delta (1+q-2q^2)+{(1-q)}^2{\varrho }^2\right)+1/2\left(\Delta (1+q)+\varrho (1-q)\right)$ and ${|}^{\kappa }{D}_q\Psi (\Delta )|=1/3\left({\Delta }^2(1+q+q^2)+\kappa \Delta (1+q-2q^2)+{(1-q)}^2{\kappa }^2\right)+1/2\left(\Delta (1+q)+\kappa (1-q)\right)$ are strongly convex functions with $w=(1+q+q^2)/3$. In Theorem 1, the left-hand side of the inequality becomes

$$\begin{array}{cc}\hfill L& :=\left|{\displaystyle \frac{\Psi \left(\kappa \right)+\Psi \left(\rho \right)}{2}}-{\displaystyle \frac{1}{2(\rho -\kappa )}}\left({\int }_{\kappa }^{\rho }\Psi {\left(\varphi \right)}_{\kappa }{d}_q\varphi +{\int }_{\kappa }^{\rho }\Psi \left(\varphi \right){}^{\rho }d_q\varphi \right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{\Psi \left(0\right)+\Psi \left(1\right)}{2}}-{\displaystyle \frac{1}{2}}\left({\int }_0^1\Psi {\left(\varphi \right)}_0{d}_q\varphi +{\int }_0^1\Psi \left(\varphi \right){}^{1}d_q\varphi \right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{5}{12}}-\left({\displaystyle \frac{5}{12}}-{\displaystyle \frac{1}{{\left[2\right]}_q}}+{\displaystyle \frac{1}{{\left[3\right]}_q}}\right)\right|\hfill \\ \hfill & =\left|{\displaystyle \frac{1}{{\left[2\right]}_q}}-{\displaystyle \frac{1}{{\left[3\right]}_q}}\right|\hfill \end{array}$$

(21)

and the right-hand side becomes

$$\begin{array}{cc}\hfill M& :={\displaystyle \frac{(\rho -\kappa )}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left(|{}^{\rho }D_q\Psi \left(\kappa \right)|+|{}_{\kappa }D_q\Psi \left(\rho \right)|\right)+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\left(|{}^{\rho }D_q\Psi \left(\rho \right)|+|{}_{\kappa }D_q\Psi \left(\kappa \right)|\right)\right)\hfill \\ \hfill & -\left({\displaystyle \frac{q^{4}w{(\rho -\kappa )}^3}{{\left[3\right]}_q{\left[4\right]}_q}}\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left({\displaystyle \frac{{\left[3\right]}_q}{3}}+{\displaystyle \frac{{\left[2\right]}_q}{2}}+{\displaystyle \frac{{(1-q)}^2}{3}}+{\displaystyle \frac{(1-q)}{2}}\right)+{\displaystyle \frac{2q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\right)-\left({\displaystyle \frac{q^4(1+q+q^2)}{3{\left[3\right]}_q{\left[4\right]}_q}}\right)\hfill \\ \hfill & ={\displaystyle \frac{5q-q^2+2q^3}{6{\left[3\right]}_q}}+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}-{\displaystyle \frac{q^4}{3{\left[4\right]}_q}}.\hfill \end{array}$$

(22)

Moreover, by setting $w=0$ in Theorem 1, the right-hand side becomes

$$\begin{array}{cc}\hfill R& :={\displaystyle \frac{(\rho -\kappa )}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left(|{}^{\rho }D_q\Psi \left(\kappa \right)|+|{}_{\kappa }D_q\Psi \left(\rho \right)|\right)+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\left(|{}^{\rho }D_q\Psi \left(\rho \right)|+|{}_{\kappa }D_q\Psi \left(\kappa \right)|\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left({\displaystyle \frac{q}{{\left[3\right]}_q}}\left({\displaystyle \frac{{\left[3\right]}_q}{3}}+{\displaystyle \frac{{\left[2\right]}_q}{2}}+{\displaystyle \frac{{(1-q)}^2}{3}}+{\displaystyle \frac{(1-q)}{2}}\right)+{\displaystyle \frac{2q^3}{{\left[2\right]}_q{\left[3\right]}_q}}\right)\hfill \\ \hfill & ={\displaystyle \frac{5q-q^2+2q^3}{6{\left[3\right]}_q}}+{\displaystyle \frac{q^3}{{\left[2\right]}_q{\left[3\right]}_q}}.\hfill \end{array}$$

(23)

By Theorem 1, (21)–(23), and $q^4/3{\left[4\right]}_q>0$, we obtain

$$\begin{array}{c}\hfill L\le M\le R.\end{array}$$

(24)

The graphical representation of inequality (24) is shown in Figure 1.

Example 4. 

Under the same setting as in Example 3, we apply Theorem 2 to $\Psi$ with $\mu =2$. The left-hand side remains unchanged, while the right-hand side is given by

$$\begin{array}{cc}\hfill M& :={\displaystyle \frac{q(\kappa -\varrho )}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+q^2{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+q^2{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_{q}w{(\kappa -\varrho )}^2}{{\left[3\right]}_q{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}})\hfill \\ \hfill & ={\displaystyle \frac{q}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q{(\frac{1}{3}(1+q+q^2)+\frac{1}{2}(1+q))}^2}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_q}{3{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q{(\frac{1}{3}{(1-q)}^2+\frac{1}{2}(1-q))}^2+4q^2}{{\left[3\right]}_q}}-{\displaystyle \frac{q^3{\left[2\right]}_q}{3{\left[4\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}).\hfill \end{array}$$

(25)

On the other hand, by setting $w=0$ in Theorem 2, the right-hand side becomes

$$\begin{array}{cc}\hfill R& :={\displaystyle \frac{q(\kappa -\varrho )}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+q^2{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+q^2{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}})\hfill \\ \hfill & ={\displaystyle \frac{q}{2{\left[2\right]}_q}}({\left({\displaystyle \frac{{\left[2\right]}_q{(\frac{1}{3}(1+q+q^2)+\frac{1}{2}(1+q))}^2}{{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{\left[2\right]}_q{(\frac{1}{3}{(1-q)}^2+\frac{1}{2}(1-q))}^2+4q^2}{{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}).\hfill \end{array}$$

(26)

By Theorem 2, (21), (25) and (26) and $q^3{\left[2\right]}_q/3{\left[4\right]}_q>0$ we obtain

$$\begin{array}{c}\hfill L\le M\le R.\end{array}$$

(27)

The graphical representation of inequality (27) is shown in Figure 2.

Example 5. 

Under the same setting as in Example 3, we apply Theorem 3 to $\Psi$ with $\mu =2$. The left-hand side remains unchanged, while the right-hand side is given by

$$\begin{array}{cc}\hfill M& :={\displaystyle \frac{q(\kappa -\varrho )}{2{\left({[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\{{\left({\displaystyle \frac{|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+q{|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+q{|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[2\right]}_q}}-{\displaystyle \frac{q^{2}w{(\kappa -\varrho )}^2}{{\left[2\right]}_q{\left[3\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \\ \hfill & ={\displaystyle \frac{q}{2{\left({\left[3\right]}_q\right)}^{\frac{1}{2}}}}\{{\left({\displaystyle \frac{{(\frac{1}{3}(1+q+q^2)+\frac{1}{2}(1+q))}^2}{{\left[2\right]}_q}}-{\displaystyle \frac{q^2}{3{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{(\frac{1}{3}{(1-q)}^2+\frac{1}{2}(1-q))}^2+4q}{{\left[2\right]}_q}}-{\displaystyle \frac{q^2}{3{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\}.\hfill \end{array}$$

(28)

For instance, by setting $w=0$ in Theorem 3, the right-hand side becomes

$$\begin{array}{cc}\hfill R& :={\displaystyle \frac{q(\kappa -\varrho )}{2{\left({[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\{{\left({\displaystyle \frac{|{\phantom{­}}_{\varrho }{D}_q{\Psi \left(\kappa \right)|}^{\mu }+q{|{\phantom{­}}_{\varrho }{D}_q\Psi \left(\varrho \right)|}^{\mu }}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{|{\phantom{­}}^{\kappa }{D}_q{\Psi \left(\varrho \right)|}^{\mu }+q{|{\phantom{­}}^{\kappa }{D}_q\Psi \left(\kappa \right)|}^{\mu }}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \\ \hfill & ={\displaystyle \frac{q}{2{\left({\left[3\right]}_q\right)}^{\frac{1}{2}}}}\{{\left({\displaystyle \frac{{(\frac{1}{3}(1+q+q^2)+\frac{1}{2}(1+q))}^2}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{{(\frac{1}{3}{(1-q)}^2+\frac{1}{2}(1-q))}^2+4q}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\}.\hfill \end{array}$$

(29)

By Theorem 3, (21), (28) and (29) and $q^2/3{\left[2\right]}_q>0$ we obtain

$$\begin{array}{c}\hfill L\le M\le R.\end{array}$$

(30)

The graphical representation of inequality (30) is shown in Figure 3.

Example 6. 

Under the same setting as in Example 3, we now apply Theorem 5 to the function $\Psi$ with $\mu =3/2$ and $\nu =3$. In this case, the left-hand side of the resulting inequality is given by

$$\begin{array}{cc}\hfill L& :=\left|\Psi \left({\displaystyle \frac{\varrho +\kappa }{2}}\right)-{\displaystyle \frac{1}{2(\rho -\kappa )}}\left({\int }_{\kappa }^{\rho }\Psi {\left(\varphi \right)}_{\kappa }{d}_q\varphi +{\int }_{\kappa }^{\rho }\Psi \left(\varphi \right){}^{\rho }d_q\varphi \right)\right|\hfill \\ \hfill & =\left|\Psi \left({\displaystyle \frac{1}{2}}\right)-{\displaystyle \frac{1}{2}}\left({\int }_0^1\Psi {\left(\varphi \right)}_0{d}_q\varphi +{\int }_0^1\Psi \left(\varphi \right){}^{1}d_q\varphi \right)\right|\hfill \\ \hfill & =\left|-{\displaystyle \frac{1}{4}}+{\displaystyle \frac{1}{{\left[2\right]}_q}}-{\displaystyle \frac{1}{{\left[3\right]}_q}}\right|\hfill \end{array}$$

(31)

and the right-hand side becomes

$$\begin{array}{cc}\hfill M& :=\left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\left\{{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}\right({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}\hfill \\ \hfill & |{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }-w{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{16{\left[2\right]}_q{\left[3\right]}_q{\left[4\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}\hfill \\ \hfill & |{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-w{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{16{\left[2\right]}_q{\left[3\right]}_q{\left[4\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =\left({\displaystyle \frac{1}{2}}\right){\left\{{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{q}{8{\left[3\right]}_q}}({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^{{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{{\left[3\right]}_q+q^2}{24{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^{{\displaystyle \frac{3}{2}}}\hfill \\ \hfill & -{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{48{\left[2\right]}_q{\left[4\right]}_q}}{)}^{{\displaystyle \frac{2}{3}}}+{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}({\displaystyle \frac{q}{8{\left[3\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^{{\displaystyle \frac{3}{2}}}\hfill \\ \hfill & +{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{2}^{{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{{\left[3\right]}_q+q^2}{24{\left[2\right]}_q}}{)}^{{\displaystyle \frac{2}{3}}}+{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}\hfill \\ \hfill & {\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^{{\displaystyle \frac{3}{2}}}+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{2}^{{\displaystyle \frac{3}{2}}}-{\displaystyle \frac{(2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q}{48{\left[2\right]}_q{\left[4\right]}_q}}{)}^{{\displaystyle \frac{2}{3}}}\}.\hfill \end{array}$$

(32)

Furthermore, by setting $w=0$ in Theorem 5, the right-hand side becomes

$$\begin{array}{cc}\hfill R& :=\left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\{{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{q}{8{\left[3\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \\ \hfill & =\left({\displaystyle \frac{1}{2}}\right){\left\{{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{q}{8{\left[3\right]}_q}}({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^{{\displaystyle \frac{3}{2}}}\right)}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^{{\displaystyle \frac{3}{2}}}\right)}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{q}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}{\left({\displaystyle \frac{q}{8{\left[3\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^{{\displaystyle \frac{3}{2}}}+{\displaystyle \frac{{\left[3\right]}_{q}q+q^3}{8{\left[2\right]}_q{\left[3\right]}_q}}{2}^{{\displaystyle \frac{3}{2}}}\right)}^{{\displaystyle \frac{2}{3}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{2-q^2}{4{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{3}}}({\displaystyle \frac{4{\left[3\right]}_q+q{\left[2\right]}_q{\left[3\right]}_q+q{\left[2\right]}_q}{8{\left[2\right]}_q{\left[3\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^{{\displaystyle \frac{3}{2}}}\hfill \\ & +{\displaystyle \frac{q{\left[3\right]}_q-1}{8{\left[2\right]}_q{\left[3\right]}_q}}{2}^{{\displaystyle \frac{3}{2}}}{)}^{{\displaystyle \frac{2}{3}}}\}.\hfill \end{array}$$

(33)

Note that $\left((2-q^2){\left[3\right]}_q{\left[4\right]}_q+(q^2-2){\left[2\right]}_q\right)/48{\left[2\right]}_q{\left[4\right]}_q>0$ and $\left({\left[3\right]}_q+q^2\right)/24{\left[2\right]}_q>0$. By Theorem 5, (31)–(33), we obtain

$$L\le M\le R.$$

(34)

The graphical representation of inequality (34) is shown in Figure 4.

Example 7. 

Under the same setting as in Example 6, we apply Theorem 6 to $\Psi$ with $\mu =\nu =2$. The left-hand side remains unchanged, while the right-hand side is given by

$$\begin{array}{cc}\hfill M& :=\left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\left\{\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right)\right({\displaystyle \frac{1}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & -w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\kappa \right)|}^{\mu }\hfill \\ \hfill & +{\displaystyle \frac{q}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\varrho \right)|}^{\mu }-{\displaystyle \frac{{\left[2\right]}_{q}qw}{8{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right)({\displaystyle \frac{1}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\varrho \right)|}^{\mu }\hfill \\ \hfill & +{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-w{\displaystyle \frac{{\left[3\right]}_q+q^2}{8{\left[2\right]}_q{\left[3\right]}_q}}{(\kappa -\varrho )}^2{)}^{{\displaystyle \frac{1}{\mu }}}+{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}\hfill \\ \hfill & {\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }-{\displaystyle \frac{{\left[2\right]}_{q}qw}{8{\left[3\right]}_q}}{(\kappa -\varrho )}^2\right)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \\ \hfill & =\left({\displaystyle \frac{1}{2}}\right)\{\left({\displaystyle \frac{q}{2{\left(2{\left[3\right]}_q\right)}^{\frac{1}{2}}}}\right){\left({\displaystyle \frac{1}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^2-{\displaystyle \frac{{\left[3\right]}_q+q^2}{24{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{q}{2}}(1+{\displaystyle \frac{1}{{\left[2\right]}_q}})+{\displaystyle \frac{q^2}{8}}(1+{\displaystyle \frac{2}{{\left[2\right]}_q}}+{\displaystyle \frac{1}{{\left[3\right]}_q}})\right)}^{{\displaystyle \frac{1}{2}}}({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^2\hfill \\ \hfill & -{\displaystyle \frac{{\left[2\right]}_{q}q}{24}}{)}^{{\displaystyle \frac{1}{2}}}+\left({\displaystyle \frac{q}{2{\left(2{\left[3\right]}_q\right)}^{\frac{1}{2}}}}\right)({\displaystyle \frac{1}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^2+{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{\left(2\right)}^2\hfill \\ \hfill & -{\displaystyle \frac{{\left[3\right]}_q+q^2}{24{\left[2\right]}_q}}{)}^{{\displaystyle \frac{1}{2}}}+{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{q}{2}}(1+{\displaystyle \frac{1}{{\left[2\right]}_q}})+{\displaystyle \frac{q^2}{8}}(1+{\displaystyle \frac{2}{{\left[2\right]}_q}}+{\displaystyle \frac{1}{{\left[3\right]}_q}})\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \times {\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^2+{\displaystyle \frac{q}{4{\left[2\right]}_q}}{\left(2\right)}^2-{\displaystyle \frac{{\left[2\right]}_{q}q}{24}}\right)}^{{\displaystyle \frac{1}{2}}}\}.\hfill \end{array}$$

(35)

Additionally, by setting $w=0$ in Theorem 6, the right-hand side becomes

$$\begin{array}{cc}\hfill R& :=\left({\displaystyle \frac{\kappa -\varrho }{2}}\right)\{\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right){\left({\displaystyle \frac{1}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}|{}_{\varrho }D_q{\Psi \left(\kappa \right)|}^{\mu }+{\displaystyle \frac{q}{4{\left[2\right]}_q}}{|{}_{\varrho }D_q\Psi \left(\varrho \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +\left({\displaystyle \frac{q}{2{\left(2{[\nu +1]}_q\right)}^{\frac{1}{\nu }}}}\right){\left({\displaystyle \frac{1}{4{\left[2\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{1+2q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\hfill \\ \hfill & +{\left({\int }_{{\displaystyle \frac{1}{2}}}^1{(1-q\varphi )}^{\nu }{d}_q\varphi \right)}^{{\displaystyle \frac{1}{\nu }}}{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}|{}^{\kappa }D_q{\Psi \left(\varrho \right)|}^{\mu }+{\displaystyle \frac{q}{4{\left[2\right]}_q}}{|{}^{\kappa }D_q\Psi \left(\kappa \right)|}^{\mu }\right)}^{{\displaystyle \frac{1}{\mu }}}\}\hfill \\ \hfill & =\left({\displaystyle \frac{1}{2}}\right)\{\left({\displaystyle \frac{q}{2{\left(2{\left[3\right]}_q\right)}^{\frac{1}{2}}}}\right){\left({\displaystyle \frac{1}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{q}{2}}(1+{\displaystyle \frac{1}{{\left[2\right]}_q}})+{\displaystyle \frac{q^2}{8}}(1+{\displaystyle \frac{2}{{\left[2\right]}_q}}+{\displaystyle \frac{1}{{\left[3\right]}_q}})\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1+q+q^2)+{\displaystyle \frac{1}{2}}(1+q)\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +\left({\displaystyle \frac{q}{2{\left(2{\left[3\right]}_q\right)}^{\frac{1}{2}}}}\right){\left({\displaystyle \frac{1}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^2+{\displaystyle \frac{1+2q}{{\left[2\right]}_q}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{q}{2}}(1+{\displaystyle \frac{1}{{\left[2\right]}_q}})+{\displaystyle \frac{q^2}{8}}(1+{\displaystyle \frac{2}{{\left[2\right]}_q}}+{\displaystyle \frac{1}{{\left[3\right]}_q}})\right)}^{{\displaystyle \frac{1}{2}}}({\displaystyle \frac{2+q}{4{\left[2\right]}_q}}{\left({\displaystyle \frac{1}{3}}(1-2q+q^2)+{\displaystyle \frac{1}{2}}(1-q)\right)}^2\hfill \\ \hfill & +{\displaystyle \frac{q}{{\left[2\right]}_q}}{)}^{{\displaystyle \frac{1}{2}}}\}.\hfill \end{array}$$

(36)

Note that $\left({\left[3\right]}_q+q^2\right)/24{\left[2\right]}_q>0$ and ${\left[2\right]}_{q}q/24>0$. By Theorem 5, (31), (35) and (36), we obtain

$$\begin{array}{c}\hfill L\le M\le R.\end{array}$$

(37)

The graphical representation of inequality (37) is shown in Figure 5.

Remark 2. 

For Examples 3–7, we constructed examples by selecting explicit functions, such as $\Psi (\Delta )={\Delta }^3/3+{\Delta }^2/2$, and verified that their q-derivatives are strongly convex. These examples were then substituted into our main results to evaluate both sides of the inequalities. The comparisons illustrate that using strongly convex functions in the framework of q-calculus provides tighter bounds than the convex case (e.g., when setting $w=0$). This process confirms that the theoretical improvements developed in this work lead to practically sharper estimates.

6. Conclusions

This paper presents refined versions of the q-Hermite–Hadamard inequality tailored to strongly convex functions by employing both left and right q-integrals. We further establish new trapezoidal- and midpoint-type inequalities for q-differentiable strongly convex functions. To validate our results, explicit examples were constructed using carefully chosen functions whose q-derivatives are strongly convex. These examples were used to evaluate and compare the proposed bounds against classical results. The numerical comparisons reveal that the derived inequalities yield sharper estimates than those based on ordinary convexity, especially when the strong convexity condition is applied. In particular, setting $w=0$ in the examples recovers the convex case, allowing direct comparison and clearly illustrating the advantage of strong convexity. These findings not only confirm the strength of our theoretical results but also open promising directions for further extensions to broader generalized convexities under the q-calculus framework.