New Approaches to Fractal–Fractional Bullen’s Inequalities Through Generalized Convexity

Abstract

This paper introduces a new identity involving fractal–fractional integrals, which allow us to derive several new Bullen-type inequalities via generalized convexity. This study provides a significant advancement in the area of fractal–fractional inequalities, presenting a range of results not only for fractional integrals and fractal calculus, but also offering a refinement of the well-known Bullen-type inequality. We further explore the connections between generalized convexity and fractal–fractional integrals, showing how the concept of generalized convexity enables the establishment of error bounds for fractal–fractional integrals involving lower-order derivatives, with an emphasis on their applications in various fields. The findings expand the current understanding of fractal–fractional inequalities and offer new insights into the use of local fractional derivatives for analyzing functions with fractional-order properties.

1. Introduction

Convexity is a fundamental concept in mathematical analysis, characterizing functions that lie below the line segment joining any two points on their graph. Formally, a function $\mathcal{L}:[{\sigma }_1,{\sigma }_2]\to \mathbb{R}$ is said to be convex if, for all ${\varkappa }_1,{\varkappa }_2\in [{\sigma }_1,{\sigma }_2]$ and $t\in [0,1]$, the inequality $\mathcal{L}(t{\varkappa }_1+(1-t){\varkappa }_2)\le t\mathcal{L}\left({\varkappa }_1\right)+(1-t)\mathcal{L}\left({\varkappa }_2\right)$ holds.

Convex functions possess a variety of interesting properties, one of the most notable being the Hermite–Hadamard inequality. This inequality provides a valuable estimation for the average value of a convex function over an interval, stating that for any convex function $\mathcal{L}$ on $[{\sigma }_1,{\sigma }_2]$, we have

$$\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\le {\displaystyle \frac{1}{{\sigma }_2-{\sigma }_1}}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right)d\varkappa \le \frac{\mathcal{L}\left({\sigma }_1\right)+\mathcal{L}\left({\sigma }_2\right)}{2}.$$

(1)

This result highlights the central role of convexity in integral inequalities and has widespread applications in both theoretical and applied fields.

In [1], Bullen established the following inequality, commonly referred to as Bullen’s inequality, for convex functions:

$${\displaystyle \frac{1}{{\sigma }_2-{\sigma }_1}}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right)d\varkappa \le \frac{1}{2}\left[\frac{\mathcal{L}\left({\sigma }_1\right)+\mathcal{L}\left({\sigma }_2\right)}{2}+\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right].$$

(2)

In [2], Tseng et al. presented the following improved version of the Hermite–Hadamard inequality by combining inequality (1) and inequality (2).

$$\begin{array}{cc}\hfill \mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\le & \frac{1}{2}\left[\mathcal{L}\left(\frac{3{\sigma }_1+{\sigma }_2}{4}\right)+\mathcal{L}\left(\frac{{\sigma }_1+3{\sigma }_2}{4}\right)\right]\hfill \\ \hfill \le & \frac{1}{{\sigma }_2-{\sigma }_1}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right)d\varkappa \hfill \\ \hfill \le & \frac{1}{2}\left[\frac{\mathcal{L}\left({\sigma }_1\right)+\mathcal{L}\left({\sigma }_2\right)}{2}+\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right]\le \frac{\mathcal{L}\left({\sigma }_1\right)+\left({\sigma }_2\right)}{2},\hfill \end{array}$$

where $\mathcal{L}$ is a convex function on $[{\sigma }_1,{\sigma }_2]$.

In [3], Xi and Qi et al. provided the following Bullen-type inequalities.

Theorem 1. 

([3]). Let $\mathcal{L}:I\to \mathbb{R}$ be a differentiable function on $I^0$ (interior of I) with $\mathcal{L}\in L_1[{\sigma }_1,{\sigma }_2]$. If $\left|{\mathcal{L}}^{\prime }\right|$ is convex on $[{\sigma }_1,{\sigma }_2]$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left[\mathcal{L}\left({\sigma }_1\right)+2\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right]-\frac{1}{{\sigma }_2-{\sigma }_1}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right)d\varkappa \right|\hfill \\ \hfill \le & \frac{{\sigma }_2-{\sigma }_1}{16}\left(\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|\right).\hfill \end{array}$$

For more results on Bullen’s inequality, we refer readers to [4,5] and the references therein.

Fractional calculus extends classical calculus by defining derivatives and integrals of arbitrary real or complex orders. This generalization allows for the modeling of processes with memory effects, long-range dependencies, and anomalous behavior, which cannot be captured by traditional integer-order models. It is widely applied in fields such as physics, engineering, and finance to describe complex systems with non-local or non-Markovian dynamics.

For specific reasons related to certain phenomena, several fractional operators have been introduced, such as the Riemann–Liouville, Hadamard, Katugampola, and others. In 2018, Sarikaya et al. introduced what are known as generalized fractional integrals in the following way:

Definition 1. 

([6]). Let $I=[{\sigma }_1,{\sigma }_2]$ and $\mathcal{L}\in L\left[I\right]$. The right-sided generalized fractional integral is given by

$${}_{{\sigma }_2^{-}}\mathcal{I}_{\varphi }\mathcal{L}\left(x\right)=\stackrel{{\sigma }_2}{\underset{x}{\int }}{\displaystyle \frac{\varphi \left(t-x\right)}{t-x}}\mathcal{L}\left(t\right)dt,x<{\sigma }_2$$

and the left-sided one is given by

$${}_{{\sigma }_1^{+}}\mathcal{I}_{\varphi }\mathcal{L}\left(x\right)=\stackrel{x}{\underset{{\sigma }_1}{\int }}{\displaystyle \frac{\varphi \left(x-t\right)}{x-t}}\mathcal{L}\left(t\right)dt,{\sigma }_1<x,$$

where ϕ satisfies

$$\underset{0}{\stackrel{1}{\int }}\frac{\varphi \left(t\right)}{t}dt<\infty .$$

What is particularly interesting about these operators is that they can reduce to several particular fractional integrals, depending on the choice of the function $\varphi$.

Indeed, for $\varphi \left(t\right)={\displaystyle \frac{t^{\alpha }}{\Gamma \left(\alpha \right)}}$ with $\alpha >0$, we recover the Riemann–Liouville fractional integrals [7]. Similarly, for $\varphi \left(t\right)={\displaystyle \frac{t^{\frac{\alpha }{k}}}{k{\Gamma }_k\left(\alpha \right)}}$ with $\alpha >0$, the k-Riemann–Liouville integrals as defined in [8] are obtained. For $\varphi \left(t\right)=t{\displaystyle \frac{{\left(log\left({\sigma }_2\right)-log({\sigma }_2-t)\right)}^{\alpha -1}}{\Gamma \left(\alpha \right)\left({\sigma }_2-t\right)}}$ with $\alpha >0$ and ${\sigma }_1\ge 1$, we recover the Hadamard’s integrals, as defined in [7], among many others [9,10,11,12,13].

In the field of fractional calculus, many researchers have investigated Bullen-type inequalities via different types of fractional integrals [14,15,16].

In [17], Zhao et al. provided the following Bullen-type inequalities via generalized fractional integrals in the following manner:

Theorem 2. 

([17]). Let $\mathcal{L}:I\to \mathbb{R}$ be a differentiable function on $I^0$ (interior of I) with $\mathcal{L}\in L_1[{\sigma }_1,{\sigma }_2]$. If $\left|{\mathcal{L}}^{\prime }\right|$ is convex on $[{\sigma }_1,{\sigma }_2]$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left(\mathcal{L}\left({\sigma }_1\right)+2\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1}{2\Delta \left(1\right)}\left({}_{{\sigma }_1^{+}}\mathcal{I}_{\varphi }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right){}_{{\sigma }_2^{-}}\mathcal{I}_{\varphi }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right)\right|\hfill \\ \hfill \le & \frac{{\sigma }_2-{\sigma }_1}{8\Delta \left(1\right)}\left[\left(2{\mathcal{W}}_1+{\mathcal{W}}_2\right)\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+\left(3{\mathcal{W}}_2-2{\mathcal{W}}_1\right)\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|\right],\hfill \end{array}$$

where

$$\Delta \left(t\right)=\underset{0}{\stackrel{t}{\int }}\frac{\varphi \left(\frac{{\sigma }_2-{\sigma }_1}{2}s\right)}{s}ds,$$

$${\mathcal{W}}_1=\underset{0}{\stackrel{1}{\int }}t\left|\frac{\Delta \left(1\right)}{2}-\Delta \left(t\right)\right|dt$$

and

$${\mathcal{W}}_2=\underset{0}{\stackrel{1}{\int }}\left|\frac{\Delta \left(1\right)}{2}-\Delta \left(t\right)\right|dt.$$

It should be noted that the above result provides a range of outcomes through various types of fractional integrals by substituting the appropriate function $\varphi$. In particular, for $\varphi \left(t\right)=t$, Theorem 2 reduces to the classical Bullen inequality given by Xi and Qi in Theorem 1.

Fractal sets are mathematical structures known for their complex, self-similar patterns, which exhibit intricate details at all scales. Due to their non-differentiable nature, these sets cannot be effectively analyzed using traditional calculus. Local fractional calculus, however, provides a framework for defining derivatives and integrals that apply specifically to functions on fractal or irregular domains. This approach is essential for modeling phenomena with irregular shapes and self-similar properties, such as those encountered in materials science, image processing, and chaotic systems, where conventional calculus is insufficient.

In [18], Yang introduced the notion of generalized convexity as follows:

Definition 2. 

([18]). Let $\mathcal{L}$ be a function defined on the interval I and taking values in the fractal ν-type set of real numbers ${\mathbb{R}}^{\nu }$. For any ${\varkappa }_1,{\varkappa }_2\in I$ and $t\in [0,1]$, if

$$\mathcal{L}\left(t{\varkappa }_1+\left(1-t\right){\varkappa }_2\right)\le t^{\nu }\mathcal{L}\left({\varkappa }_1\right)+{\left(1-t\right)}^{\nu }\mathcal{L}\left({\varkappa }_2\right)$$

holds, then $\mathcal{L}$ is a generalized convex function on I.

In the exploration of fractal inequalities, several researchers have proposed new estimates associated with different quadrature formulas and generalized convexity forms; see [19,20,21,22,23,24,25,26,27,28] and the references therein.

In [29], Yu et al. introduced a novel class of generalized integrals, termed fractal–fractional integrals, defined as follows:

Definition 3. 

The left fractal–fractional integral is defined by

$${}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left(x\right)={\displaystyle \frac{1}{\Gamma (1+\nu )}}{\int }_{{\sigma }_1}^x{\displaystyle \frac{\varphi \left(x-t\right)}{{\left(x-t\right)}^{\nu }}}\mathcal{L}\left(t\right){\left(dt\right)}^{\nu },{\sigma }_1<x,$$

(3)

and the right-sided one is given by

$${}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left(x\right)={\displaystyle \frac{1}{\Gamma (1+\nu )}}{\int }_x^{{\sigma }_2}{\displaystyle \frac{\varphi \left(t-x\right)}{{\left(t-x\right)}^{\nu }}}\mathcal{L}\left(t\right){\left(dt\right)}^{\nu },\varkappa <{\sigma }_2,$$

(4)

where the function ϕ meets the following condition:

$${\displaystyle \frac{1}{\Gamma (1+\nu )}}{\int }_0^1{\displaystyle \frac{\varphi \left(t\right)}{t^{\nu }}}{\left(dt\right)}^{\nu }<{\infty }^{\nu }.$$

In the same study [29], they derived midpoint-type inequalities for functions that are twice differentiable in the local fractional sense and exhibit generalized convexity. Expanding on this, Butt et al. investigated Simpson’s inequality in [30] and proposed parametric inequalities in [31] for functions with generalized convex second-order local fractional derivatives. Later, in [32], Yuan et al. introduced the class of generalized $(P,m)$-convex functions and established a multi-parameter identity involving fractal–fractional integral operators, leading to interesting results for twice-differentiable generalized $(P,m)$-convex functions under local fractional calculus. More recently, Alsharari et al. [33] provided the fractal–fractional version of Simpson-type inequalities via generalized s-convexity.

In [34], Butt et al. established the following fractal–fractional Bullen-type inequalities for generalized convex functions.

Theorem 3. 

([34]). Let $\mathcal{L}:I\to {\mathbb{R}}^{\nu }$ be a local fractional differentiable function on $I^{\circ }$, ${\sigma }_1,{\sigma }_2\in I^{\circ }$ with ${\sigma }_1<{\sigma }_2$, such that ${\mathcal{L}}^{\left(\nu \right)}$ is local fractional continuous on $\left[{\sigma }_1,{\sigma }_2\right]$. If $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$ is generalized convex on $\left[{\sigma }_1,{\sigma }_2\right]$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+{}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\Delta }_{\nu }\left(1\right)}\left[\left(2{\mathcal{W}}_1^{\nu }+{\mathcal{W}}_2^{\nu }\right)\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+\left(3{\mathcal{W}}_2^{\nu }-{\mathcal{W}}_1^{\nu }\right)\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right],\hfill \end{array}$$

where

$${\Delta }_{\nu }\left(t\right)=\frac{1}{\Gamma (1+\nu )}\underset{0}{\stackrel{t}{\int }}\frac{\varphi \left(\frac{{\sigma }_2-{\sigma }_1}{2}s\right)}{s^{\nu }}{\left(ds\right)}^{\nu },$$

$${\mathcal{W}}_1^{\nu }=\frac{1}{\Gamma (1+\nu )}\underset{0}{\stackrel{1}{\int }}{t}^{\nu }\left|\frac{{\Delta }_{\nu }\left(1\right)}{2^{\nu }}-{\Delta }_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }$$

and

$${\mathcal{W}}_2^{\nu }=\frac{1}{\Gamma (1+\nu )}\underset{0}{\stackrel{1}{\int }}\left|\frac{{\Delta }_{\nu }\left(1\right)}{2^{\nu }}-{\Delta }_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }.$$

Here, it is worth noting that, for $\nu =1$, the result obtained in Theorem 3 coincides with that of Theorem 2.

In this paper, we present a new identity involving fractal–fractional integrals, from which we derive several novel Bullen-type inequalities using the generalized convexity of first-order local fractional derivatives. While the result obtained by Butt et al. in Theorem 3 from [34] represents the fractal version of the result established by Zhao et al. in Theorem 2 from [17], which in turn reduces to the result obtained by Xi and Qi in Theorem 1 from [3] for $\varphi \left(t\right)=t$, the approach followed in this study provides a new fractal–fractional version of Bullen-type inequalities, which not only provides a wide range of results for fractional integrals and fractal calculus, but also refines the result obtained by Xi and Qi in [3], highlighting the versatility of this framework.

The article is structured as follows: In Section 2, we recall some concepts from local fractional calculus that are essential for the subsequent analysis. Section 3 introduces a new fractal–fractional identity involving first-order local fractional derivatives, which serves as the foundation for the main result established in Section 4. This section also includes a numerical example with a graphical representation to compare the proposed result with that of Theorem 1 from [3], demonstrating that the former represents an improvement. Additional results derived using the improved generalized Hölder’s inequality and the improved generalized power mean inequality are presented in Section 5. Section 6 is dedicated to showcasing some applications of the obtained results. Finally, the study concludes with a summary presented in Section 7.

2. Preliminaries on Local Fractional Calculus

In this section, we introduce some fundamental concepts from fractal theory. For $0<\nu \le 1$, the following $\nu$-type sets are defined:

The $\nu$-type set of integers is given by:

$${\mathbb{Z}}^{\nu }:=\left\{0^{\nu },\pm 1^{\nu },\pm 2^{\nu },\dots ,\pm n^{\nu },\dots \right\}.$$

The $\nu$-type set of rational numbers is:

$${\mathbb{Q}}^{\nu }:=\left\{m^{\nu }={\left(\frac{u}{v}\right)}^{\nu }:u,v\in \mathbb{Z},v\ne 0\right\}.$$

The $\nu$-type set of irrational numbers is:

$${\mathbb{J}}^{\nu }:=\left\{m^{\nu }\ne {\left(\frac{u}{v}\right)}^{\nu }:u,v\in \mathbb{Z},v\ne 0\right\}.$$

The fractal set of real numbers is defined by:

$${\mathbb{R}}^{\nu }:={\mathbb{Q}}^{\nu }\cup {\mathbb{J}}^{\nu }.$$

If $u_1^{\nu },u_2^{\nu },$ and $u_3^{\nu }$ belong to ${\mathbb{R}}^{\nu }$, then the following properties are true:

1. $u_1^{\nu }+u_2^{\nu }$ and $u_1^{\nu }{u}_2^{\nu }$ belong to ${\mathbb{R}}^{\nu }$. 2. $u_1^{\nu }+u_2^{\nu }=u_2^{\nu }+u_1^{\nu }={(u+v)}^{\nu }={(v+u)}^{\nu }$. 3. $u_1^{\nu }+(u_2^{\nu }+u_3^{\nu })={(u+v)}^{\nu }+u_3^{\nu }$. 4. $u_1^{\nu }{u}_2^{\nu }=u_2^{\nu }{u}_1^{\nu }={\left(uv\right)}^{\nu }={\left(vu\right)}^{\nu }$. 5. $u_1^{\nu }\left(u_2^{\nu }{u}_3^{\nu }\right)=\left(u_1^{\nu }{u}_2^{\nu }\right)u_3^{\nu }$. 6. $u_1^{\nu }(u_2^{\nu }+u_3^{\nu })=u_1^{\nu }{u}_2^{\nu }+u_1^{\nu }{u}_3^{\nu }$. 7. $u_1^{\nu }+0^{\nu }=0^{\nu }+u_1^{\nu }=u_1^{\nu }$ and $u_1^{\nu }{1}^{\nu }=1^{\nu }{u}_1^{\nu }=u_1^{\nu }$.

Definition 4. 

([18]). A function $\mathcal{L}:\mathbb{R}\to {\mathbb{R}}^{\nu }$ is said to be local fractional continuous at $y_0$ if

$$\forall ϵ>0,\exists \delta >0:\left|\mathcal{L}\left(y\right)-\mathcal{L}\left(y_0\right)\right|<{ϵ}^{\nu }$$

whenever $|y-y_0|<\delta$, with $ϵ,\delta \in \mathbb{R}$. The set of all locally fractional continuous functions on $({\sigma }_1,{\sigma }_2)$ is denoted by $C_{\nu }({\sigma }_1,{\sigma }_2)$.

Definition 5. 

([18]). The local fractional derivative of $\mathcal{L}\left(y\right)$ of order ν at $y=y_0$ is defined as:

$${\mathcal{L}}^{\left(\nu \right)}\left(y_0\right)={\left.{\displaystyle \frac{d^{\nu }\mathcal{L}\left(y\right)}{d{y}^{\nu }}}\right|}_{y=y_0}=\underset{y\to y_0}{lim}{\displaystyle \frac{{\Delta }^{\nu }(\mathcal{L}\left(y\right)-\mathcal{L}\left(y_0\right))}{{(y-y_0)}^{\nu }}},$$

where ${\Delta }^{\nu }(\mathcal{L}\left(y\right)-\mathcal{L}\left(y_0\right))\cong \Gamma (\nu +1)(\mathcal{L}\left(y\right)-\mathcal{L}\left(y_0\right))$.

If there exists a function ${\mathcal{L}}^{\left(k\nu \right)}\left(y\right)=\stackrel{k\mathrm{times}}{\overbrace{D^{\nu }{D}^{\nu }\dots D^{\nu }}}\mathcal{L}\left(y\right)$ for all $y\in I\subseteq \mathbb{R}$, we say that $\mathcal{L}\in D_{k\nu }\left(I\right)$, where $k=0,1,2,3,\dots$.

Definition 6.

([18]). Let $\mathcal{L}\left(y\right)\in C_{\nu }[{\sigma }_1,{\sigma }_2]$. The local fractional integral of $\mathcal{L}$ is given by:

$${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right)={\displaystyle \frac{1}{\Gamma (\nu +1)}}{\int }_{{\sigma }_1}^{{\sigma }_2}\mathcal{L}\left(y\right){\left(dy\right)}^{\nu }={\displaystyle \frac{1}{\Gamma (\nu +1)}}\underset{\Delta y\to 0}{lim}\sum _{j=0}^{N-1}\mathcal{L}\left(y_j\right){(\Delta y_j)}^{\nu },$$

where $\Delta y_j=y_{j+1}-y_j$ and $\Delta y=max\left\{\Delta y_1,\Delta y_2,\dots ,\Delta y_{N-1}\right\}$, with the partition $[y_0,y_1,\dots ,y_N]$ of the interval $[{\sigma }_1,{\sigma }_2]$.

Thus, we have ${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right)=0$ if ${\sigma }_1={\sigma }_2$, and ${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right)=-{}_{{\sigma }_2}I_{{\sigma }_1}^{\nu }\mathcal{L}\left(y\right)$ if ${\sigma }_1<{\sigma }_2$. If for any $y\in [{\sigma }_1,{\sigma }_2]$, there exists ${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right)$, we denote this as $\mathcal{L}\left(y\right)\in I_y^{\nu }[{\sigma }_1,{\sigma }_2]$.

Lemma 1. 

([18]). If $\mathcal{L}\left(y\right)={\mathcal{A}}^{\left(\nu \right)}\left(y\right)\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$, then we have

$${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right)=\mathcal{A}\left({\sigma }_2\right)-\mathcal{A}\left({\sigma }_1\right).$$

Lemma 2. 

([18]). If $\mathcal{L},\mathcal{A}\in D_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$ and ${\mathcal{L}}^{\left(\nu \right)}\left(y\right),{\mathcal{A}}^{\left(\nu \right)}\left(y\right)\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$, then we have

$${}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }\mathcal{L}\left(y\right){\mathcal{A}}^{\left(\nu \right)}\left(y\right)={\left.\mathcal{L}\left(y\right)\mathcal{A}\left(y\right)\right|}_{{\sigma }_1}^{{\sigma }_2}-{}_{{\sigma }_1}I_{{\sigma }_2}^{\nu }{\mathcal{L}}^{\left(\nu \right)}\left(y\right)\mathcal{A}\left(y\right).$$

Lemma 3. 

([18]). For $\mathcal{L}\left(y\right)=y^{k\nu }$, we have

$$\begin{array}{cc}\hfill \frac{d^{\nu }{y}^{k\nu }}{d{y}^{\nu }}=& \frac{\Gamma \left(1+k\nu \right)}{\Gamma \left(1+\left(k-1\right)\nu \right)}{y}^{\left(k-1\right)\nu },\hfill \\ \hfill \frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{y}^{k\nu }{\left(dy\right)}^{\nu }=& \frac{\Gamma \left(1+k\nu \right)}{\Gamma \left(1+\left(k+1\right)\nu \right)}\left({\sigma }_2^{\left(k+1\right)\nu }-{\sigma }_1^{\left(k+1\right)\nu }\right),k\in \mathbb{R}.\hfill \end{array}$$

In the work of Yang [18], the generalized Hölder inequality in the context of fractal domains is presented as follows:

Lemma 4. 

([18]). Let $\mathcal{L},\mathcal{A}\in C_{\nu }[{\sigma }_1,{\sigma }_2]$, and ${\mathcal{L}}^p$, ${\mathcal{A}}^q$ are local fractional integral on $[{\sigma }_1,{\sigma }_2]$ for $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then

$$\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\mathcal{A}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\le {\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{\left|\mathcal{L}\left(\varkappa \right)\right|}^p{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}.$$

Recently, Luo et al. [35] presented an improved form of the generalized Hölder integral inequality.

Lemma 5. 

([35]). Let $\mathcal{L},\mathcal{A}\in C_{\nu }[{\sigma }_1,{\sigma }_2]$, and ${\mathcal{L}}^p$, ${\mathcal{A}}^q$ are local fractional integral on $[{\sigma }_1,{\sigma }_2]$ for $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then

$$\begin{array}{cc}\hfill & \frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\mathcal{A}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\hfill \\ \hfill \le & {\left(\frac{1}{{\sigma }_2-{\sigma }_1}\right)}^{\nu }\left[{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{({\sigma }_2-\varkappa )}^{\nu }{\left|\mathcal{L}\left(\varkappa \right)\right|}^p{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{({\sigma }_2-\varkappa )}^{\nu }{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{(\varkappa -{\sigma }_1)}^{\nu }{\left|\mathcal{L}\left(\varkappa \right)\right|}^p{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{(\varkappa -{\sigma }_1)}^{\nu }{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Yu et al. [36] presented the generalized power mean inequality in the context of fractal domains as follows:

Lemma 6. 

([36]). Let $\mathcal{L},\mathcal{A}\in C_{\nu }[{\sigma }_1,{\sigma }_2]$, and $\mathcal{L}$, $\mathcal{L}{\mathcal{A}}^q$ are local fractional integral on $[{\sigma }_1,{\sigma }_2]$ for $q\ge 1$, then

$$\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\mathcal{A}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\le {\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\right|{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}.$$

In the same study, they proposed an improved version of generalized power mean inequality in the following manner:

Lemma 7. 

([36]). Let $\mathcal{L},\mathcal{A}\in C_{\nu }[{\sigma }_1,{\sigma }_2]$, and $\mathcal{L}$, $\mathcal{L}{\mathcal{A}}^q$ are local fractional integral on $[{\sigma }_1,{\sigma }_2]$ for $q\ge 1$, then

$$\begin{array}{cc}\hfill & \frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\left|\mathcal{L}\left(\varkappa \right)\mathcal{A}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\hfill \\ \hfill \le & {\left(\frac{1}{{\sigma }_2-{\sigma }_1}\right)}^{\nu }\left[{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{({\sigma }_2-\varkappa )}^{\nu }\left|\mathcal{L}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{({\sigma }_2-\varkappa )}^{\nu }\left|\mathcal{L}\left(\varkappa \right)\right|{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{(\varkappa -{\sigma }_1)}^{\nu }\left|\mathcal{L}\left(\varkappa \right)\right|{\left(d\varkappa \right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}{(\varkappa -{\sigma }_1)}^{\nu }\left|\mathcal{L}\left(\varkappa \right)\right|{\left|\mathcal{A}\left(\varkappa \right)\right|}^q{\left(d\varkappa \right)}^{\nu }\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

3. Auxiliary Result

In this section, we introduce a new fractal–fractional identity involving first-order local fractional derivatives, which forms the basis for the main results of this study.

We assume that $\varphi :\left[0,\infty \right)\to \left[0^{\nu },{\infty }^{\nu }\right)$ satisfy the following condition:

$$\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)s\right)}{s^{\nu }}{\left(ds\right)}^{\nu }<{\infty }^{\nu }.$$

Lemma 8. 

Let $\mathcal{L}:I\to {\mathbb{R}}^{\nu }$ be a differentiable function on $I^{\circ }$, ${\sigma }_1,{\sigma }_2\in I^{\circ }$ with ${\sigma }_1<{\sigma }_2$, and ${\mathcal{L}}^{\left(\nu \right)}\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$; then, the following equality holds

$$\begin{array}{cc}\hfill & \frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\hfill \\ \hfill =& \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{4^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill & -\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill & -\left.\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }\right),\hfill \end{array}$$

where

$${\mathcal{M}}_{\nu }\left(t\right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{t}{\underset{0}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)s\right)}{s^{\nu }}{\left(ds\right)}^{\nu }<{\infty }^{\nu }$$

and

$${\mathcal{N}}_{\nu }\left(t\right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{t}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)s\right)}{s^{\nu }}{\left(ds\right)}^{\nu }<{\infty }^{\nu }.$$

Proof. 

Let

$$\mathcal{K}=2^{\nu }{\mathcal{K}}_1+{\mathcal{K}}_2-2^{\nu }{\mathcal{K}}_3-{\mathcal{K}}_4,$$

(5)

where

$$\begin{array}{cc}\hfill {\mathcal{K}}_1=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu },\hfill \\ \hfill {\mathcal{K}}_2=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu },\hfill \\ \hfill {\mathcal{K}}_3=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu },\hfill \\ \hfill {\mathcal{K}}_4=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }.\hfill \end{array}$$

Using Lemma 2 for ${\mathcal{K}}_1$, we obtain

$$\begin{array}{cc}\hfill {\mathcal{K}}_1=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& {\left.\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(t\right)\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}_0^{\frac{1}{2}}\hfill \\ \hfill & -\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& \frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(\frac{1}{2}\right)\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)-\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }.\hfill \end{array}$$

(6)

Similarly, we obtain for ${\mathcal{K}}_2$

$$\begin{array}{cc}\hfill {\mathcal{K}}_2=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& {\left.\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right)\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}_{\frac{1}{2}}^1\hfill \\ \hfill & -\frac{2^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& \frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(1\right)\mathcal{L}\left({\sigma }_2\right)-\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}\left({\mathcal{M}}_{\nu }\left(\frac{1}{2}\right)-{\mathcal{N}}_{\nu }\left(\frac{1}{2}\right)\right)\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\hfill \\ \hfill & -\frac{2^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& \frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(1\right)\mathcal{L}\left({\sigma }_2\right)+\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}\left({\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(\frac{1}{2}\right)\right)\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\hfill \\ \hfill & -\frac{2^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right){\left(dt\right)}^{\nu }.\hfill \end{array}$$

(7)

Likewise, we obtain

$$\begin{array}{cc}\hfill {\mathcal{K}}_3=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\mathcal{M}}_{\nu }\left(t\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& -\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(\frac{1}{2}\right)\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill {\mathcal{K}}_4=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left({\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right){\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }\hfill \\ \hfill =& -\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{\mathcal{M}}_{\nu }\left(1\right)\mathcal{L}\left({\sigma }_1\right)-\frac{1^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}\left({\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(\frac{1}{2}\right)\right)\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\hfill \\ \hfill & +\frac{2^{\nu }}{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\frac{\varphi \left(\left({\sigma }_2-{\sigma }_1\right)t\right)}{t^{\nu }}\mathcal{L}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right){\left(dt\right)}^{\nu }.\hfill \end{array}$$

(9)

By substituting (6)–(9) into (5), and subsequently multiplying both sides of the resulting equation by ${\displaystyle \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{4^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}}$, we obtain the desired result. □

4. Primary Findings

In this section, we establish the main result based on the fractal–fractional identity introduced earlier and provide a numerical example with a graphical representation to validate and compare our findings with those of Theorem 1.

Theorem 4. 

Let $\mathcal{L}:I\to {\mathbb{R}}^{\nu }$ be a differentiable function on $I^{\circ }$, ${\sigma }_1,{\sigma }_2\in I^{\circ }$ with ${\sigma }_1<{\sigma }_2$, such that $\mathcal{L}\in D_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$ and ${\mathcal{L}}^{\left(\nu \right)}\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$. If $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$ is generalized convex on $\left[{\sigma }_1,{\sigma }_2\right]$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({\mathcal{R}}_1\left(\nu \right)\left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right)+{\mathcal{R}}_2\left(\nu \right)\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right),\hfill \end{array}$$

where

$${\mathcal{R}}_1\left(\nu \right)=\frac{1^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right){\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }$$

(10)

and

$${\mathcal{R}}_2\left(\nu \right)=\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right)t^{\nu }{\left(dt\right)}^{\nu }.$$

(11)

Proof. 

From Lemma 8, along with the modulus properties and the fact that $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$ is generalized convex, it follows that

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{4^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\hfill \\ \hfill & +\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\hfill \\ \hfill & +\left.\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\right)\hfill \\ \hfill =& \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|{\left(dt\right)}^{\nu }\right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)-{\mathcal{N}}_{\nu }\left(1-\frac{t}{2}\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(t\frac{{\sigma }_1+{\sigma }_2}{2}+\left(1-t\right){\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\hfill \\ \hfill & +\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(t\frac{{\sigma }_1+{\sigma }_2}{2}+\left(1-t\right){\sigma }_2\right)\right|{\left(dt\right)}^{\nu }\hfill \\ \hfill & +\left.\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)-{\mathcal{N}}_{\nu }\left(1-\frac{t}{2}\right)\right|\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|{\left(dt\right)}^{\nu }\right)\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\left({\left(1-t\right)}^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+t^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right){\left(dt\right)}^{\nu }\right.\hfill \\ \hfill & +\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|\left({\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)-{\mathcal{N}}_{\nu }\left(1-\frac{t}{2}\right)\right)\right|\left(t^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|+{\left(1-t\right)}^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right){\left(dt\right)}^{\nu }\hfill \\ \hfill & +\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\left(t^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|+{\left(1-t\right)}^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right){\left(dt\right)}^{\nu }\hfill \\ \hfill & +\left.\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left|{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)-{\mathcal{N}}_{\nu }\left(1-\frac{t}{2}\right)\right|\left({\left(1-t\right)}^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+t^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right){\left(dt\right)}^{\nu }\right)\hfill \\ \hfill =& \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right)t^{\nu }{\left(dt\right)}^{\nu }\times \left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right.\hfill \\ \hfill & \left.+\frac{1^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right){\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }\times \left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right)\right)\hfill \\ \hfill =& \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({\mathcal{R}}_1\left(\nu \right)\left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right)+{\mathcal{R}}_2\left(\nu \right)\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right),\hfill \end{array}$$

where we have used (10), (11), and the fact that ${\mathcal{M}}_{\nu }\left(t\right)+{\mathcal{N}}_{\nu }\left(t\right)={\mathcal{M}}_{\nu }\left(1\right)$. The proof is completed. □

Corollary 1. 

By making use of the generalized convexity of $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$ (i.e.: $\left|{\mathcal{L}}^{\prime }\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\le \frac{\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|}{2}$), Theorem 4 gives

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}{\mathcal{R}}_3\left(\nu \right)\left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|\right),\hfill \end{array}$$

where

$${\mathcal{R}}_3\left(\nu \right)=\frac{1^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right){\left(dt\right)}^{\nu }.$$

(12)

Corollary 2. 

If we attempt to take $\nu =1$ in Theorem 4, then we obtain the following Bullen’s inequality via generalized fractional integrals

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left(\mathcal{L}\left({\sigma }_1\right)+2\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1}{2{\mathcal{M}}_1\left(1\right)}\left({}_{{\sigma }_2^{-}}\mathcal{I}_{\varphi }\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}\mathcal{I}_{\varphi }\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{\left({\sigma }_2-{\sigma }_1\right)}{8{\mathcal{M}}_1\left(1\right)}\left({\mathcal{R}}_1\left(1\right)\left(\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|\right)+{\mathcal{R}}_2\left(1\right)\left|{\mathcal{L}}^{\prime }\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|\right),\hfill \end{array}$$

where ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ are defined as in (10) and (11), respectively.

Moreover, using the generalized convexity of $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left(\mathcal{L}\left({\sigma }_1\right)+2\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1}{2{\mathcal{M}}_1\left(1\right)}\left({}_{{\sigma }_2^{-}}\mathcal{I}_{\varphi }\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}\mathcal{I}_{\varphi }\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{\left({\sigma }_2-{\sigma }_1\right)}{8{\mathcal{M}}_1\left(1\right)}{\mathcal{R}}_3\left(1\right)\left(\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|\right),\hfill \end{array}$$

where ${\mathcal{R}}_3$ is defined as in (12).

Corollary 3. 

In Corollary 2, if we attempt to set $\varphi \left(t\right)=t$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1}{4}\left[\mathcal{L}\left({\sigma }_1\right)+2\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right]-\frac{1}{{\sigma }_2-{\sigma }_1}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right)d\varkappa \right|\hfill \\ \hfill \le & \frac{{\sigma }_2-{\sigma }_1}{32}\left(\left|{\mathcal{L}}^{\prime }\left({\sigma }_1\right)\right|+2\left|{\mathcal{L}}^{\prime }\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|+\left|{\mathcal{L}}^{\prime }\left({\sigma }_2\right)\right|\right).\hfill \end{array}$$

The above result provides a more refined and therefore improved estimation compared to that given by Xi and Qi in Theorem 1, as their result can be derived by simply using the convexity of $\left|{\mathcal{L}}^{\prime }\right|$ for the term $\left|{\mathcal{L}}^{\prime }\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)\right|$.

Example 1. 

Consider the function $\mathcal{L}:[{\sigma }_1,{\sigma }_2]\to \mathbb{R}$ defined by $\mathcal{L}\left(\varkappa \right)={\varkappa }^3$, with $0<{\sigma }_1<{\sigma }_2$.

The results presented in Theorem 1 and Corollary 3 related to the function $\mathcal{L}$ under consideration are given as follows:

$$\left|\frac{1}{4}\left({\sigma }_1^3+\frac{{\left({\sigma }_1+{\sigma }_2\right)}^3}{4}+{\sigma }_2^3\right)-\frac{\left({\sigma }_1+{\sigma }_1\right)\left({\sigma }_1^2+{\sigma }_2^2\right)}{4}\right|\le \frac{3\left({\sigma }_2-{\sigma }_1\right)}{16}\left({\sigma }_1^2+{\sigma }_2^2\right)$$

(13)

and

$$\left|\frac{1}{4}\left({\sigma }_1^3+\frac{{\left({\sigma }_1+{\sigma }_2\right)}^3}{4}+{\sigma }_2^3\right)-\frac{\left({\sigma }_1+{\sigma }_1\right)\left({\sigma }_1^2+{\sigma }_2^2\right)}{4}\right|\le \frac{3\left({\sigma }_2-{\sigma }_1\right)}{64}\left(3{\sigma }_1+2{\sigma }_1{\sigma }_2+3{\sigma }_2\right).$$

(14)

For $\left({\sigma }_1,{\sigma }_2\right)\in \left([1,2]\times [2,3]\right)$, the graphical representation of results (13) and (14) is shown in Figure 1, where the left-hand side (LHS) is depicted in red, and the right-hand sides of the two inequalities (13) (RHS1) and (14) (RHS2) are represented in blue and green, respectively.

From the representations in Figure 1, it is clear that the result in Corollary 3 is significantly better than the one given by Xi and Qi in Theorem 1.

5. Further Results

In this section, we present additional results derived using the improved generalized Hölder’s inequality and the improved generalized power mean inequality, further enriching the analysis.

Theorem 5. 

Let $\mathcal{L}:I\to {\mathbb{R}}^{\nu }$ be a differentiable function on $I^{\circ }$, ${\sigma }_1,{\sigma }_2\in I^{\circ }$ with ${\sigma }_1<{\sigma }_2$, such that $\mathcal{L}\in D_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$ and ${\mathcal{L}}^{\left(\nu \right)}\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$. If ${\left|{\mathcal{L}}^{\left(\nu \right)}\right|}^q$ is generalized convex on $\left[{\sigma }_1,{\sigma }_2\right]$, where $q>1$ with $\frac{1}{p}+\frac{1}{q}=1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left[\left({\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}+2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right)\right.\hfill \\ \hfill & \times \left({\left({\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill & +\left({\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}+2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left.\left.\left({\left({\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right)\right)\right],\hfill \end{array}$$

where ${\mathcal{Q}}_1\left(\nu \right),{\mathcal{Q}}_2\left(\nu \right),{\mathcal{Q}}_3\left(\nu \right)$ and ${\mathcal{Q}}_4\left(\nu \right)$ are defined as in (15)–(18), respectively.

Proof. 

Using Lemma 8, the improved generalized Hölder’s inequality, and the fact that ${\left|{\mathcal{L}}^{\left(\nu \right)}\right|}^q$ is generalized convex, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +\left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}{\left({\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill =& \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left[\left({\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}+2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right)\right.\hfill \\ \hfill & \times \left({\left({\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{Q}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill & +\left({\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}+2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left|{\mathcal{M}}_{\nu }\left(t\right)\right|}^p{\left(dt\right)}^{\nu }\right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left.\left.\left({\left({\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{Q}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{Q}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right)\right)\right],\hfill \end{array}$$

where the following has been applied:

$$\begin{array}{cc}\hfill {\mathcal{Q}}_1\left(\nu \right)=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{t}^{\nu }{\left(dt\right)}^{\nu }\hfill \\ \hfill =& {\left(\frac{7}{8}\right)}^{\nu }\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}-{\left(\frac{3}{8}\right)}^{\nu }\frac{\Gamma \left(1+\nu \right)}{\Gamma \left(1+2\nu \right)},\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_2\left(\nu \right)=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }{t}^{\nu }{\left(dt\right)}^{\nu }=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }{\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }\hfill \\ \hfill =& {\left(\frac{1}{8}\right)}^{\nu }\left(\frac{\Gamma \left(1+\nu \right)}{\Gamma \left(1+2\nu \right)}-\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill {\mathcal{Q}}_3\left(\nu \right)=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }{\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }{t}^{\nu }{\left(dt\right)}^{\nu }\hfill \\ \hfill =& {\left(\frac{1}{4}\right)}^{\nu }\frac{\Gamma \left(1+\nu \right)}{\Gamma \left(1+2\nu \right)}-{\left(\frac{1}{8}\right)}^{\nu }\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}\hfill \end{array}$$

(17)

and

$$\begin{array}{cc}\hfill {\mathcal{Q}}_4\left(\nu \right)=& \frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{2\nu }{\left(dt\right)}^{\nu }=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{2\nu }{\left(dt\right)}^{\nu }={\left(\frac{1}{8}\right)}^{\nu }\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}.\hfill \end{array}$$

(18)

The proof is completed. □

Theorem 6. 

Let $\mathcal{L}:I\to {\mathbb{R}}^{\nu }$ be a differentiable function on $I^{\circ }$, ${\sigma }_1,{\sigma }_2\in I^{\circ }$ with ${\sigma }_1<{\sigma }_2$, such that $\mathcal{L}\in D_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$ and ${\mathcal{L}}^{\left(\nu \right)}\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$. If ${\left|{\mathcal{L}}^{\left(\nu \right)}\right|}^q$ is generalized convex on $\left[{\sigma }_1,{\sigma }_2\right]$, where $q\ge 1$, then we have

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\right.{\left({\mathcal{S}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_5\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_6\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_7\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_8\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_6\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_5\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\left.{\left({\mathcal{S}}_8\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_7\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where ${\mathcal{S}}_i$ are given by (19)–(26), respectively.

Proof. 

Using Lemma 8, the improved generalized power mean inequality, and the fact that ${\left|{\mathcal{L}}^{\left(\nu \right)}\right|}^q$ is generalized convex, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(\left(1-t\right){\sigma }_1+t{\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left|{\mathcal{L}}^{\left(\nu \right)}\left(t{\sigma }_1+\left(1-t\right){\sigma }_2\right)\right|}^q{\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\right)\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\right.\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left({\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times {\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\hfill \\ \hfill & \times \left.{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|\left(t^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\left(1-t\right)}^{\nu }{\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right){\left(dt\right)}^{\nu }\right)}^{\frac{1}{q}}\right).\hfill \end{array}$$

So, we have

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\sigma }_1\right)+2^{\nu }\mathcal{L}\left(\frac{{\sigma }_1+{\sigma }_2}{2}\right)+\mathcal{L}\left({\sigma }_2\right)\right)-\frac{1^{\nu }}{2^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left({}_{{\sigma }_2^{-}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_1\right)+{}_{{\sigma }_1^{+}}{}^{\varphi }\mathcal{I}^{\left(\nu \right)}\mathcal{L}\left({\sigma }_2\right)\right)\right|\hfill \\ \hfill \le & \frac{{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}{8^{\nu }{\mathcal{M}}_{\nu }\left(1\right)}\left(2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\right.{\left({\mathcal{S}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_5\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_6\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_7\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_8\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_2\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_1\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +2^{\nu }{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_4\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_3\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}{\left({\mathcal{S}}_6\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_5\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ \hfill & +{\left(\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(dt\right)}^{\nu }\right)}^{1-\frac{1}{q}}\left.{\left({\mathcal{S}}_8\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_1\right)\right|}^q+{\mathcal{S}}_7\left(\nu \right){\left|{\mathcal{L}}^{\left(\nu \right)}\left({\sigma }_2\right)\right|}^q\right)}^{\frac{1}{q}}\right),\hfill \end{array}$$

where the following has been applied:

$${\mathcal{S}}_1\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(1-t\right)}^{s\nu }{\left(dt\right)}^{\nu },$$

(19)

$${\mathcal{S}}_2\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{\left(\frac{1}{2}-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|t^{s\nu }{\left(dt\right)}^{\nu },$$

(20)

$${\mathcal{S}}_3\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}{t}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)\right|{\left(1-t\right)}^{s\nu }{\left(dt\right)}^{\nu },$$

(21)

$${\mathcal{S}}_4\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{\frac{1}{2}}{\underset{0}{\int }}\left|{\mathcal{M}}_1\left(t\right)\right|t^{\left(1+s\right)\nu }{\left(dt\right)}^{\nu },$$

(22)

$${\mathcal{S}}_5\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(1-t\right)}^{2\nu }{\left(dt\right)}^{\nu },$$

(23)

$${\mathcal{S}}_6\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(1-t\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|t^{\nu }{\left(dt\right)}^{\nu },$$

(24)

$${\mathcal{S}}_7\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|{\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }$$

(25)

and

$${\mathcal{S}}_8\left(\nu \right)=\frac{1}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{\frac{1}{2}}{\int }}{\left(t-\frac{1}{2}\right)}^{\nu }\left|{\mathcal{M}}_{\nu }\left(t\right)-{\mathcal{N}}_{\nu }\left(t\right)\right|t^{\nu }{\left(dt\right)}^{\nu }.$$

(26)

The proof is completed. □

6. Applications

In this section, we explore several applications of the obtained results, highlighting their practical relevance and potential implications.

6.1. Quadrature Formula

Let $\mathrm{Y}$ be the partition of the points ${\sigma }_1={\varkappa }_0<{\varkappa }_1<\cdots <{\varkappa }_n={\sigma }_2$ of the interval $\left[{\sigma }_1,{\sigma }_2\right]$, and consider the quadrature formula

$$\frac{1}{\Gamma \left(\nu +1\right)}\underset{{\sigma }_1}{\stackrel{{\sigma }_2}{\int }}\mathcal{L}\left(\varkappa \right){\left(d\varkappa \right)}^{\nu }=\mathsf{\Lambda }\left(\mathcal{L},\mathrm{Y}\right)+R\left(\mathcal{L},\mathrm{Y}\right),$$

where

$$\mathsf{\Lambda }\left(\mathcal{L},\mathrm{Y}\right)=\frac{1}{\Gamma \left(\nu +1\right)}\underset{i=0}{\sum ^{n-1}}\frac{{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\varkappa }_i\right)+2^{\nu }\mathcal{L}\left(\frac{{\varkappa }_i+{\varkappa }_{i+1}}{2}\right)+\mathcal{L}\left({\varkappa }_{i+1}\right)\right)$$

and $R\left(\mathcal{L},\mathrm{Y}\right)$ represents the corresponding approximation error.

Proposition 1. 

Let $n\in \mathbb{N}$ and $\mathcal{L}:\left[{\sigma }_1,{\sigma }_2\right]\to {\mathbb{R}}^{\nu }$ be a differentiable function on $\left[{\sigma }_1,{\sigma }_2\right]$ with $0\le {\sigma }_1<{\sigma }_2$ and ${\mathcal{L}}^{\left(\nu \right)}$$\in C_{\nu }\left[{\sigma }_1,{\sigma }_2\right]$. If $\left|{\mathcal{L}}^{\left(\nu \right)}\right|$ is generalized convex on $\left[{\sigma }_1,{\sigma }_2\right]$, we have

$$\begin{array}{cc}\hfill & \left|R\left(\mathcal{L},\mathrm{Y}\right)\right|\hfill \\ \hfill \le & \frac{1}{\Gamma \left(1+\nu \right)}\underset{i=0}{\sum ^{n-1}}\frac{{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{2\nu }}{4^{\nu }}\left(\left(\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}+{\left(\frac{1}{3}\right)}^{\nu }\frac{\Gamma \left(1+\nu \right)}{\Gamma \left(1+2\nu \right)}\right)\left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\varkappa }_i\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\varkappa }_{i+1}\right)\right|\right)\right.\hfill \\ \hfill & +\left.2^{\nu }\left({\left(\frac{4}{3}\right)}^{\nu }\frac{\Gamma \left(1+\nu \right)}{\Gamma \left(1+2\nu \right)}-\frac{\Gamma \left(1+2\nu \right)}{\Gamma \left(1+3\nu \right)}\right)\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\varkappa }_i+{\varkappa }_{i+1}}{2}\right)\right|\right).\hfill \end{array}$$

Proof. 

Applying Theorem 4 with $\varphi \left(t\right)=t^{\nu }$ on the subintervals $\left[{\varkappa }_i,{\varkappa }_{i+1}\right]$$\left(i=0,1,...,n-1\right)$ of the partition $\mathrm{Y}$, we obtain

$$\begin{array}{cc}\hfill & \left|\frac{1^{\nu }}{4^{\nu }}\left(\mathcal{L}\left({\varkappa }_i\right)+2^{\nu }\mathcal{L}\left(\frac{{\varkappa }_i+{\varkappa }_{i+1}}{2}\right)+\mathcal{L}\left({\varkappa }_{i+1}\right)\right)-\frac{\Gamma \left(\nu +1\right)}{{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{}_{{\varkappa }_i}I_{{\varkappa }_{i+1}}^{\nu }\mathcal{L}\left(t\right)\right|\hfill \\ \hfill \le & \frac{\Gamma \left(1+\nu \right){\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{8^{\nu }\Gamma \left(2\nu \right)}\left(\left|{\mathcal{L}}^{\left(\nu \right)}\left({\varkappa }_i\right)\right|+\left|{\mathcal{L}}^{\left(\nu \right)}\left({\varkappa }_{i+1}\right)\right|+2^{\nu }\left|{\mathcal{L}}^{\left(\nu \right)}\left(\frac{{\varkappa }_i+{\varkappa }_{i+1}}{2}\right)\right|\right),\hfill \end{array}$$

(27)

where we have used the facts that

$${\mathcal{M}}_{\nu }\left(t\right)=\frac{{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{\Gamma \left(1+\nu \right)}{t}^{\nu },$$

(28)

$${\mathcal{R}}_1\left(\nu \right)=\frac{1^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right){\left(1-t\right)}^{\nu }{\left(dt\right)}^{\nu }=\frac{{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{\Gamma \left(1+2\nu \right)}$$

(29)

and

$${\mathcal{R}}_2\left(\nu \right)=\frac{2^{\nu }}{\Gamma \left(1+\nu \right)}\stackrel{1}{\underset{0}{\int }}\left(\left|{\mathcal{M}}_{\nu }\left(1\right)-2^{\nu }{\mathcal{M}}_{\nu }\left(1-\frac{t}{2}\right)\right|+2^{\nu }\left|{\mathcal{M}}_{\nu }\left(\frac{t}{2}\right)\right|\right)t^{\nu }{\left(dt\right)}^{\nu }=\frac{2^{\nu }{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }}{\Gamma \left(1+2\nu \right)}.$$

(30)

Multiplying both sides of inequality (27) by ${\displaystyle \frac{1}{\Gamma \left(1+\nu \right)}}{\left({\varkappa }_{i+1}-{\varkappa }_i\right)}^{\nu }$, summing the inequalities over $i=0$ to $n-1$, and using the triangle inequality leads to the desired result. □

6.2. Application to Special Means

The generalized Arithmetic mean and is defined as:

$$A_{\nu }\left({\sigma }_1,{\sigma }_2\right)={\displaystyle \frac{{\sigma }_1^{\nu }+{\sigma }_2^{\nu }}{2^{\nu }}}.$$

And the generalized p-Logarithmic mean:

$${\mathcal{L}}_{p\nu }\left({\sigma }_1,{\sigma }_2\right)={\left[{\displaystyle \frac{\Gamma \left(1+p\nu \right)}{\Gamma \left(1+\left(p+1\right)\nu \right)}}\left({\displaystyle \frac{{\sigma }_2^{\left(p+1\right)\nu }-{\sigma }_1^{\left(p+1\right)\nu }}{(p+1){\left({\sigma }_2-{\sigma }_1\right)}^{\nu }}}\right)\right]}^{\frac{1}{p}},$$

for ${\sigma }_1,{\sigma }_2\in \mathbb{R}$, with ${\sigma }_1\ne {\sigma }_2$ and $p\in \mathbb{Z}‵\left\{0,-1\right\}$.

Proposition 2. 

Let ${\sigma }_1,{\sigma }_2\in \mathbb{R}$ with $0<{\sigma }_1<{\sigma }_2$, and $n\in \mathbb{N}$ with $n\ge 2$; then, we have

$$\begin{array}{cc}\hfill & \left|A_{\nu }\left({\sigma }_1^n,{\sigma }_2^n\right)+A_{\nu }^n\left({\sigma }_1,{\sigma }_2\right)-2^{\nu }{L}_{n\nu }^n\left({\sigma }_1,{\sigma }_2\right)\right|\hfill \\ \hfill \le & \frac{\left(1^{\nu }+2^{\left(n-2\right)\nu }\right)\Gamma \left(1+\nu \right)\Gamma \left(1+n\nu \right)}{2^{n\nu }\Gamma \left(1+2\nu \right)\Gamma \left(1+\left(n-1\right)\nu \right)}{\left({\sigma }_2-{\sigma }_1\right)}^{\nu }\left({\sigma }_1^{\left(n-1\right)\nu }+{\sigma }_2^{\left(n-1\right)\nu }\right).\hfill \end{array}$$

Proof. 

The conclusion is derived from Theorem 4 with $\varphi \left(t\right)=t^{\nu }$, applied to the function $\mathfrak{h}\left(\varkappa \right)={\varkappa }^{n\nu }$, utilizing the relations (27)–(30). □

7. Conclusions

In this study, we have presented a new fractal–fractional version of Bullen-type inequalities, which refines and extends existing results. The main findings address the research problem by offering a novel approach to fractal–fractional analysis, providing a deeper understanding of the relationships between local fractional derivatives and fractal functions. Our results demonstrate improvements over classical inequalities, offering a more generalized framework with potential applications in various fields.

However, several open research issues remain. Future work could explore further generalizations of the inequalities presented here, considering other types of generalized convexity. Additionally, while our approach contributes to the current body of knowledge, limitations arise for other types of inequalities and formulas. A more in-depth exploration of these formulas would be valuable for expanding the scope of the results and addressing the specific needs of various applications in science and engineering. Critically, some key studies have not fully explored the implications of combining this domain, which presents an opportunity for future investigations to address these gaps and refine the current understanding of fractal–fractional inequalities.