On Proportional Caputo-Hybrid Fractional Milne-Type Inequalities: Theory, Numerical Simulations, and Applications

Abstract

The goal of this study is to establish a new type of Milne-type inequality in the scope of fractional calculus with the aid of proportional Caputo-hybrid operators. We will focus on two different scopes of regularity, which contain functions whose first and second derivatives are convex, and functions whose first and second derivatives are Lipschitz continuous. We will base these estimates on a new integral identity of proportional Caputo-hybrid integrals. We will show that the smoothness of the derivative influences the shape of the bounds. Convexity will cause symmetry. Lipschitz continuity will contain bounds on the modulus of continuity. To show that our results are accurate and easy to obtain, we included a full numerical example with graphics and applications to quadrature error estimation.

1. Introduction

Numerical integration without a primitive can be daunting. However, numerical quadrature provides a solution to this problem by approximating the integral value through a weighted sum of the integral’s values at pre-chosen points (or nodes). The trapezoidal and Simpson’s rules offer a foundation for various modern numerical integration techniques. However, the aforementioned rules’ accuracy depends on the smoothness and shape of the function being integrated.

Numerous publications have focused on analyzing the precision of these methods, particularly when the integrated function does not possess a high degree of smoothness. Among the most successful functions to analyze smoothness is the convex function. Many researchers have successfully used various degrees of convexity (classical, strong, generalized) to form inequalities that bound the differences between the true value of an integral and its numerical approximation. For a pertinent review see [1].

Definition 1

([2]). A function $\mathcal{F}:\mathbf{I}\subset \mathbb{R}\to \mathbb{R}$ is deemed to be convex if

$$\mathcal{F}\left(\mu \mathfrak{v}+\left(1-\mu \right)\mathfrak{w}\right)\le \mu \mathcal{F}\left(\mathfrak{v}\right)+\left(1-\mu \right)\mathcal{F}\left(\mathfrak{w}\right)$$

is valid for all $\mathfrak{v},\mathfrak{w}\in \mathbf{I}$ and all $\mu \in [0,1]$.

One of the most interesting extensions of the convexity notion is s-convexity in the second sense, which is formulated as follows.

Definition 2

([3]). A nonnegative function $\mathcal{F}:\mathbf{I}\subset {\mathbb{R}}^{+}=\left[0,\infty \right)\to \mathbb{R}$ is deemed to be s-convex in the second sense for some fixed $s\in \left(0,1\right]$ if

$$\mathcal{F}(\mu \mathfrak{v}+(1-\mu )\mathfrak{w})\le {\mu }^s\mathcal{F}\left(\mathfrak{v}\right)+{(1-\mu )}^s\mathcal{F}\left(\mathfrak{w}\right)$$

is valid for all $\mathfrak{v},\mathfrak{w}\in \mathbf{I}$ and $\mu \in [0,1]$.

Among the increasingly studied rules is the Milne-type quadrature, which shares the three-point structure of Simpson’s formula but incorporates a distinct weighting that yields different error characteristics and is given by

$$\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)={\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\simeq {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi .$$

(1)

This rule has recently drawn attention due to its favorable convergence properties and its utility in constructing refined integral inequalities. Several authors have explored Milne-type inequalities under various convexity frameworks, including [4,5,6,7,8], where new estimates and generalizations have been established in both classical and extended settings.

Parallel to these developments, fractional calculus has provided a fertile ground for generalizing integral inequalities by introducing non-local operators that capture memory and hereditary effects. Fractional variants of Milne-type inequalities have been investigated using Riemann–Liouville fractional integrals [9,10,11,12], Ktugampola fractional integrals [13,14], and conformable fractional integrals [15,16], revealing deeper connections between numerical analysis and fractional modeling. For further results on Milne-type inequalities via different types of integral operators, we refer the reader to [17,18,19].

In [10], Budak and Hyder established the following Milne-type inequality for differentiable convex functions via Riemann–Liouville fractional integrals.

Theorem 1

([10]). Let $\mathcal{F}:[{\mathfrak{d}}_1,{\mathfrak{d}}_2]\to \mathbb{R}$ be a differentiable mapping on $({\mathfrak{d}}_1,{\mathfrak{d}}_2)$ such that $\mathcal{F}\in L_1\left([{\mathfrak{d}}_1,{\mathfrak{d}}_2]\right)$, where $L_1\left([{\mathfrak{d}}_1,{\mathfrak{d}}_2]\right)$ denotes the class of measurable functions that are absolutely integrable with ${\displaystyle {\int }_{{\sigma }_1}^{{\sigma }_2}\left|\mathcal{F}\left(t\right)\right|\mathrm{d}t<\infty }$. If $\left|{\mathcal{F}}^{\prime }\right|$ is convex on $[{\mathfrak{d}}_1,{\mathfrak{d}}_2]$. Then, we have

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\Gamma \left(\varrho +1\right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }}}\left({\mathbf{J}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)+{\mathbf{J}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{2}}\left({\mathcal{B}}_1\left(\varrho \right)+{\mathcal{B}}_2\left(\varrho \right)\right)\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where ${\mathbf{J}}_{{\mathfrak{d}}_1^{+}}^{\varrho }$ and ${\mathbf{J}}_{{\mathfrak{d}}_2^{-}}^{\varrho }$ denote the left- and right-hand-sided Riemann–Liouville fractional integrals of order ϱ expressed as

$${\mathbf{J}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\varrho \right)}}{\int }_{{\mathfrak{d}}_1}^x{\left(x-\mu \right)}^{\varrho -1}\mathcal{F}\left(\mu \right)\mathrm{d}\mu ,x>{\mathfrak{d}}_1,$$

$${\mathbf{J}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\varrho \right)}}{\int }_x^{{\mathfrak{d}}_2}{\left(\mu -x\right)}^{\varrho -1}\mathcal{F}\left(\mu \right)\mathrm{d}\mu ,x<{\mathfrak{d}}_2,$$

where $\Gamma \left(\varrho \right)$ denotes the Euler Gamma function $\Gamma \left(\varrho \right)=\underset{0}{\stackrel{1}{\int }}{\mu }^{\varrho -1}{e}^{-\mu }\mathrm{d}\mu$, and ${\mathcal{B}}_1\left(\varrho \right)$ and ${\mathcal{B}}_2\left(\varrho \right)$ are given by

$${\mathcal{B}}_1\left(\varrho \right)=\left\{\begin{array}{ccc}{\displaystyle \frac{2\varrho }{\varrho +1}}{\left({\displaystyle \frac{2}{3}}\right)}^{\frac{1}{\varrho }+1}+{\displaystyle \frac{1}{2^{\varrho +1}\left(\varrho +1\right)}}-{\displaystyle \frac{1}{3}}\hfill & for& 0<\varrho \le {\displaystyle \frac{\ln 3-\ln 2}{\ln 2}}\\ {\displaystyle \frac{1}{3}}-{\displaystyle \frac{1}{2^{\varrho +1}\left(\varrho +1\right)}}\hfill & for& \varrho >{\displaystyle \frac{\ln 3-\ln 2}{\ln 2}}\end{array}\right.$$

and

$${\mathcal{B}}_2\left(\varrho \right)=\left\{\begin{array}{ccc}{\displaystyle \frac{1}{\varrho +1}}-{\displaystyle \frac{1}{2^{\varrho +1}\left(\varrho +1\right)}}-\frac{1}{6}\hfill & for& 0<\varrho \le {\displaystyle \frac{\ln 3}{\ln 2}}\\ {\displaystyle \frac{2\varrho }{\varrho +1}}{\left({\displaystyle \frac{1}{3}}\right)}^{\frac{1}{\varrho }+1}+{\displaystyle \frac{1}{2^{\varrho +1}\left(\varrho +1\right)}}+{\displaystyle \frac{1}{\varrho +1}}-{\displaystyle \frac{1}{2}}\hfill & for& \varrho >{\displaystyle \frac{\ln 3}{\ln 2}},\end{array}\right.$$

respectively.

In [20], Baleanu et al. introduced a novel fractional operator based on a generalized proportional derivative. This operator admits an equivalent representation as a Riemann–Liouville integral applied to the proportional derivative. Moreover, under certain key specializations, it reduces to a linear combination of a Riemann–Liouville fractional integral and a Caputo fractional derivative.

Definition 3

([20]). Let $\mathcal{F}:\mathbf{I}\subset {\mathbb{R}}^{+}\to \mathbb{R}$ be a differentiable function on ${\mathbf{I}}^{\circ }$ (the interior of the interval $\mathbf{I}$), and $\mathcal{F},{\mathcal{F}}^{\prime }\in L\left(\mathbf{I}\right)$. The proportional Caputo-hybrid operator of order ϱ is defined as

$${}_0\mathcal{D}_{\varpi }^{\varrho }\mathcal{F}\left(\varpi \right)={\displaystyle \frac{1}{\Gamma \left(1-\varrho \right)}}\stackrel{\varpi }{\underset{0}{\int }}\left({\mathcal{K}}_1\left(\varrho ,\mathfrak{v}\right)\mathcal{F}\left(\mathfrak{v}\right)+{\mathcal{K}}_0\left(\varrho ,\mathfrak{v}\right){\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\right){\left(\varpi -\mathfrak{v}\right)}^{-\varrho }\mathrm{d}\mathfrak{v},$$

where $\varrho \in \left(0,1\right)$ and ${\mathcal{K}}_0$ and ${\mathcal{K}}_1$ are two variable functions $(0,1)\times \mathbf{I}\to \mathbb{R}$ and satisfy

$$\underset{\varrho \to 0^{+}}{\lim }{\mathcal{K}}_0(\varrho ,\mathfrak{v})=0;\underset{\varrho \to 1^{-}}{\lim }{\mathcal{K}}_0(\varrho ,\mathfrak{v})=1;with{\mathcal{K}}_0(\varrho ,\mathfrak{v})\ne 0,$$

and

$$\underset{\varrho \to 0^{+}}{\lim }{\mathcal{K}}_1(\varrho ,\mathfrak{v})=0;\underset{\varrho \to 1^{-}}{\lim }{\mathcal{K}}_1(\varrho ,\mathfrak{v})=0;with{\mathcal{K}}_1(\varrho ,\mathfrak{v})\ne 0.$$

Remark 1

([20]). By assigning the boundary values $\varrho =0$ and $\varrho =1$, we deduce the following distinct forms:

$$\underset{\varrho \to 0}{\lim }{}_0\mathcal{D}_{\varpi }^{\varrho }\mathcal{F}\left(\varpi \right)=\underset{0}{\stackrel{\varpi }{\int }}\mathcal{F}\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v},$$

and

$$\underset{\varrho \to 1}{\lim }{}_0\mathcal{D}_{\varpi }^{\varrho }\mathcal{F}\left(\varpi \right)={\mathcal{F}}^{\prime }\left(\varpi \right).$$

Subsequently, by using specific functions ${\mathcal{K}}_0$ and ${\mathcal{K}}_1$, Sarikaya presented the following definition of proportional Caputo-hybrid fractional operators:

Definition 4

([21]). Let $\mathcal{F}:\mathbf{I}\subset {\mathbb{R}}^{+}\to \mathbb{R}$ be a differentiable function on ${\mathbf{I}}^{\circ }$ (the interior of the interval $\mathbf{I}$), and $\mathcal{F},{\mathcal{F}}^{\prime }\in L_1\left(\mathbf{I}\right)$. The left- and right-sided proportional Caputo-hybrid operators of order ϱ are defined, respectively, as follows:

$${\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)={\displaystyle \frac{1}{\Gamma \left(1-\varrho \right)}}\stackrel{{\mathfrak{d}}_2}{\underset{{\mathfrak{d}}_1}{\int }}\left({\mathcal{K}}_1\left(\varrho ,{\mathfrak{d}}_2-\mathfrak{v}\right)\mathcal{F}\left(\mathfrak{v}\right)+{\mathcal{K}}_0\left(\varrho ,{\mathfrak{d}}_2-\mathfrak{v}\right){\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\right){\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{-\varrho }\mathrm{d}\mathfrak{v},$$

and

$${\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)={\displaystyle \frac{1}{\Gamma \left(1-\varrho \right)}}\stackrel{{\mathfrak{d}}_2}{\underset{{\mathfrak{d}}_1}{\int }}\left({\mathcal{K}}_1\left(\varrho ,\mathfrak{v}-{\mathfrak{d}}_1\right)\mathcal{F}\left(\mathfrak{v}\right)+{\mathcal{K}}_0\left(\varrho ,\mathfrak{v}-{\mathfrak{d}}_1\right){\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\right){\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{-\varrho }\mathrm{d}\mathfrak{v},$$

where $\varrho \in \left(0,1\right)$ and ${\mathcal{K}}_0\left(\varrho ,\mathfrak{v}\right)={(1-\varrho )}^2{\mathfrak{v}}^{1-\varrho }$ and ${\mathcal{K}}_1\left(\varrho ,\mathfrak{v}\right)={\varrho }^2{\mathfrak{v}}^{\varrho }$.

We emphasize that, consistent with the observations in Remark 1, choosing the values $\varrho =0$ and $\varrho =1$ results in

$$\underset{\varrho \to 0}{\lim }{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)=\underset{\varrho \to 1}{\lim }{\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)=\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v},$$

and

$$\underset{\varrho \to 1}{\lim }{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)={\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right),\underset{\varrho \to 0}{\lim }{\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)={\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right).$$

For the limiting case $\varrho \to 1$, the convergence of the singular terms to the Dirac delta distribution $\delta ({\mathfrak{d}}_2-\mathfrak{v})$ and $\delta (\mathfrak{v}-{\mathfrak{d}}_1)$ act as an approximation to the identity, which, coupled with the vanishing kernel ${\mathcal{K}}_1$, directly extracts the first derivative ${\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)$ and ${\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)$ from the integral.

In the same work [21], Sarikaya presented the Hermite–Hadamard and trapezium-type inequalities using convexity via proportional Caputo-hybrid operators. Furthermore, via the same operators and for functions whose first-order and second-order derivatives are convex, Sarikaya [22] gave the Simpson-type inequality. In [23], Demir and Tunç established Newton-type inequalities for twice differentiable functions via proportional Caputo-hybrid operators. More recently, Mehtab et al. [24] provided corrected Euler–Maclaurin-type inequalities via convexity, along with computational analysis and applications.

In [25], Demir established the following Milne-type inequalities for functions whose first-order and second-order derivatives are convex and $\varrho \in (0,1)$:

$$\begin{array}{cc}\hfill & |{\displaystyle \frac{{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }}{2^{\varrho }}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}{2^{2-\varrho }}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2^{\varrho }{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)}^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)}^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)|\hfill \\ \hfill \le & {\displaystyle \frac{5{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho +1}}{3\times 2^{\varrho +3}}}\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right)\hfill \\ \hfill & +\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }\left({\displaystyle \frac{2^{\varrho -3}\left(11-8\varrho \right)}{6(3-2\varrho )}}\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where $\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)$ is defined as in (1).

In an additional publication [26], the author derived inequalities of the Milne type for several categories of functions, such as bounded mappings, Lipschitz continuous functions, and those of bounded variation.

In this work, we establish new variants of Milne-type inequalities via proportional Caputo-hybrid operators, leveraging the convexity and Lipschitz continuity of ${\mathcal{F}}^{\prime }$ and ${\mathcal{F}}^{″}$. To this end, we begin by introducing a novel integral identity involving proportional Caputo-hybrid integrals in Section 2. Building on this identity, we derive numerous Milne-type inequalities in Section 3 for functions whose first-order and second-order derivatives are either convex or Lipschitz continuous. To validate the sharpness and highlight the practical implications of our results, in Section 4 we provide a numerical example with graphical illustration, along with some applications. The study concludes with a summary of the main findings in Section 5.

2. Proportional Caputo-Hybrid Identity

To lay the analytical groundwork for our inequalities, we first establish a new integral identity involving proportional Caputo-hybrid fractional integrals.

Lemma 1.

Let $\mathcal{F}:\mathbf{I}\to \mathbb{R}$ be a twice differentiable function on ${\mathbf{I}}^{\circ }$, where ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in {\mathbf{I}}^{\circ }$ satisfying ${\mathfrak{d}}_1<{\mathfrak{d}}_2$ and let $\mathcal{F},{\mathcal{F}}^{\prime },{\mathcal{F}}^{″}\in L_1\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$. Then, for $\varrho \in (0,1)$, the following equality

$$\begin{array}{cc}\hfill & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left(\mu -{\displaystyle \frac{2}{3}}\right)\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right)\left({\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right)\left({\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \right),\hfill \end{array}$$

holds, where $\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)$ is defined as in (1), and $\Gamma (·)$ is referred to as the well-known Γ-function.

Proof. 

Let

$${\mathcal{I}}_1=\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left(\mu -{\displaystyle \frac{2}{3}}\right)\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu ,$$

$${\mathcal{I}}_2=\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right)\left({\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu$$

and

$${\mathcal{I}}_3=\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right)\left({\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu .$$

Using the integration by parts, ${\mathcal{I}}_1$ gives

$$\begin{array}{cc}\hfill {\mathcal{I}}_1=& {\left.{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left(\mu -{\displaystyle \frac{2}{3}}\right)\left(\mathcal{F}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+\mathcal{F}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\right|}_0^{\frac{1}{2}}\hfill \\ \hfill & -{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left(\mathcal{F}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+\mathcal{F}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left[{\displaystyle \frac{2}{3}}\left(\mathcal{F}\left({\mathfrak{d}}_1\right)+\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{3}}\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left(\mathcal{F}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+\mathcal{F}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu .\hfill \end{array}$$

(2)

Likewise, we get

$$\begin{array}{cc}\hfill {\mathcal{I}}_2=& {\left.{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right)\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\right|}_0^{\frac{1}{2}}\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\mu }^{1-2\varrho }\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left[2\left({\left({\displaystyle \frac{1}{2}}\right)}^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right){\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+{\displaystyle \frac{2}{3}}\left({\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)+{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\mu }^{1-2\varrho }\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu ,\hfill \end{array}$$

(3)

and

$$\begin{array}{cc}\hfill {\mathcal{I}}_3=& {\left.{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right)\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\right|}_{\frac{1}{2}}^1\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\mu }^{1-2\varrho }\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left[{\displaystyle \frac{2}{3}}\left({\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)+{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-\left({\left({\displaystyle \frac{1}{2}}\right)}^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right){\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)\right]\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\displaystyle \frac{1}{2}}}{\stackrel{1}{\int }}{\mu }^{1-2\varrho }\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu .\hfill \end{array}$$

(4)

By performing a change of variables, (2) yields

$$\begin{array}{cc}\hfill {\mathcal{I}}_1=& {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\displaystyle \frac{2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)}{3}}\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{1}{\int }}\mathcal{F}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\mathrm{d}\mu \hfill \\ \hfill =& {\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\displaystyle \frac{2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)}{3}}\right)-{\displaystyle \frac{1}{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^2}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}.\hfill \end{array}$$

(5)

Adding (3) and (4), then using change of variables, we obtain

$$\begin{array}{cc}\hfill {\mathcal{I}}_2+{\mathcal{I}}_3=& {\displaystyle \frac{2}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\displaystyle \frac{2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)}{3}}\right)\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{0}{\stackrel{1}{\int }}{\mu }^{1-2\varrho }\left({\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)+{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right)\mathrm{d}\mu \hfill \\ \hfill =& {\displaystyle \frac{2}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\left({\displaystyle \frac{2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)}{3}}\right)\hfill \\ \hfill & -{\displaystyle \frac{2-2\varrho }{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{3-2\varrho }}}\left(\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}{\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{1-2\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}+\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}{\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{1-2\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}\right).\hfill \end{array}$$

(6)

Multiplying (5) by ${\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }$, and (6) by ${\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}$, and summing the resulting equalities, we obtain

$$\begin{array}{cc}\hfill & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }{\mathcal{I}}_1+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\mathcal{I}}_2+{\mathcal{I}}_3\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\left({\displaystyle \frac{2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)}{3}}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\left({\displaystyle \frac{2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)}{3}}\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varrho }^2}{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}-{\displaystyle \frac{{\left(1-\varrho \right)}^2}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left(\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}{\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{1-2\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}+\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}{\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{1-2\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)\mathrm{d}\mathfrak{v}\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\left({\displaystyle \frac{2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)}{3}}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\left({\displaystyle \frac{2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)}{3}}\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left(\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\left({\left(1-\varrho \right)}^2{\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{1-\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)+{\varrho }^2{\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{F}\left(\mathfrak{v}\right)\right){\left(\mathfrak{v}-{\mathfrak{d}}_1\right)}^{-\varrho }\mathrm{d}\mathfrak{v}\right.\hfill \\ \hfill & +\left.\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\left({\left(1-\varrho \right)}^2{\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{1-\varrho }{\mathcal{F}}^{\prime }\left(\mathfrak{v}\right)+{\varrho }^2{\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{\varrho }\mathcal{F}\left(\mathfrak{v}\right)\right){\left({\mathfrak{d}}_2-\mathfrak{v}\right)}^{-\varrho }\mathrm{d}\mathfrak{v}\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\left({\displaystyle \frac{2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)}{3}}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\left({\displaystyle \frac{2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left(\frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)}{3}}\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right).\hfill \end{array}$$

Thus we have proved the claim. □

3. Proportional Caputo-Hybrid Milne-Type Inequalities

Armed with the identity provided in Lemma 1, we derive Milne-type inequalities under two distinct regularity assumptions. First, when the first-order and second-order derivatives of the underlying function are convex; second, when these derivatives satisfy Lipschitz continuity conditions.

Theorem 2.

Let $\mathcal{F}:\mathbf{I}\subset {\mathbb{R}}^{+}\to \mathbb{R}$ be a twice differentiable function on ${\mathbf{I}}^{\circ }$, where ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in {\mathbf{I}}^{\circ }$ satisfying ${\mathfrak{d}}_1<{\mathfrak{d}}_2$, and let $\mathcal{F},{\mathcal{F}}^{\prime },{\mathcal{F}}^{″}\in L_1\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$. If $\left|{\mathcal{F}}^{\prime }\right|$ and $\left|{\mathcal{F}}^{″}\right|$ are s-convex on $\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$, then the following inequality holds for $\varrho \in (0,1)$:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{C}\left(s\right)\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right)+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\mathcal{D}}_1\left(\varrho ,s\right)+{\mathcal{D}}_2\left(\varrho ,s\right)\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where $\mathcal{C}$, ${\mathcal{D}}_1$, and ${\mathcal{D}}_2$, are defined as in (7), (8), and (9), respectively.

Proof. 

From Lemma 1, the s-convexity of $\left|{\mathcal{F}}^{\prime }\right|$ and $\left|{\mathcal{F}}^{″}\right|$, we have the next absolute values inequality:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left(\left|{\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|+\left|{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left(\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|+\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left(\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|+\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \right)\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left({\left(1-\mu \right)}^s+{\mu }^s\right)\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left({\left(1-\mu \right)}^s+{\mu }^s\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \right.\hfill \\ \hfill & +\left.\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left({\left(1-\mu \right)}^s+{\mu }^s\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right)\mathrm{d}\mu \right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{C}\left(s\right)\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right)+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\mathcal{D}}_1\left(\varrho ,s\right)+{\mathcal{D}}_2\left(\varrho ,s\right)\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where we have used

$$\mathcal{C}\left(s\right)=\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left({\left(1-\mu \right)}^s+{\mu }^s\right)\mathrm{d}\mu ={\displaystyle \frac{1}{(s+1)(s+2)}}\left({\displaystyle \frac{2s+1}{3}}+{\left({\displaystyle \frac{1}{2}}\right)}^{s+1}\right),$$

(7)

$$\begin{array}{ccc}\hfill {\mathcal{D}}_1\left(\varrho ,s\right)& =\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left({\mu }^s+{\left(1-\mu \right)}^s\right)\mathrm{d}\mu \hfill & \\ & =\left\{\begin{array}{ccc}\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\displaystyle \frac{2}{3}}-{\mu }^{2-2\varrho }\right)\left({\mu }^s+{\left(1-\mu \right)}^s\right)\mathrm{d}\mu \hfill & \hspace{1em}\hspace{1em} \mathrm{if}& 0<\varrho \le {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\\ \underset{0}{\stackrel{{\left(\frac{2}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}{\int }}\left({\displaystyle \frac{2}{3}}-{\mu }^{2-2\varrho }\right)\left({\mu }^s+{\left(1-\mu \right)}^s\right)\mathrm{d}\mu +\underset{{\left(\frac{2}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}{\stackrel{\frac{1}{2}}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right)\left({\mu }^s+{\left(1-\mu \right)}^s\right)\mathrm{d}\mu \hfill & \hspace{1em}\hspace{1em} \mathrm{if}& \varrho >{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\end{array}\right.\hfill \\ & =\left\{\begin{array}{ccc}{\displaystyle \frac{2}{3(s+1)}}-{\displaystyle \frac{2^{2\varrho -s-3}}{s+3-2\varrho }}-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right)\hfill & \mathrm{if}& 0<\varrho \le {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\\ -2\left[{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{{\left(\frac{2}{3}\right)}^{\frac{s+1}{2-2\varrho }}+1-{\left(1-{\left(\frac{2}{3}\right)}^{\frac{1}{2-2\varrho }}\right)}^{s+1}}{s+1}}\right)-\left({\displaystyle \frac{{\left(\frac{2}{3}\right)}^{\frac{s+3-2\varrho }{2-2\varrho }}}{s+3-2\varrho }}+{\mathbf{B}}_{{\left(\frac{2}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,s+1\right)\right)\right],\hfill & \mathrm{if}& \varrho >{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}}\end{array}\right.\hfill \end{array}$$

(8)

and

$$\begin{array}{cc}\hfill {\mathcal{D}}_2\left(\varrho ,s\right)=& \underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\displaystyle \frac{1}{3}}-{\mu }^{2-2\varrho }\right|\left({\mu }^s+{(1-\mu )}^s\right)\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}\underset{\frac{1}{2}}{\stackrel{{\left(\frac{1}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right)\left({\mu }^s+{(1-\mu )}^s\right)\mathrm{d}\mu \hfill & \hspace{1em}\hspace{1em} \mathrm{if}& 0<\varrho \le {\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\\ \underset{{\left(\frac{1}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}{\stackrel{1}{\int }}\left({\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right)\left({\mu }^s+{(1-\mu )}^s\right)\mathrm{d}\mu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\displaystyle \frac{1}{3}}-{\mu }^{2-2\varrho }\right|\left({\mu }^s+{(1-\mu )}^s\right)\mathrm{d}\mu \hfill & \hspace{1em}\hspace{1em} \mathrm{if}& \varrho >{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\end{array}\right.\hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \frac{1-2^{-(s+3-2\varrho )}}{s+3-2\varrho }}+\mathbf{B}(3-2\varrho ,s+1)-{\displaystyle \frac{1}{3(s+1)}}\hfill & & \\ -2\left[{\displaystyle \frac{{\left(\frac{1}{3}\right)}^{\frac{s+3-2\varrho }{2-2\varrho }}-{\left(\frac{1}{2}\right)}^{s+3-2\varrho }}{s+3-2\varrho }}+{\mathbf{B}}_{{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,s+1\right)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right)\right]\hfill & & \\ +{\displaystyle \frac{2}{3}}\left({\displaystyle \frac{{\left(\frac{1}{3}\right)}^{\frac{s+1}{2-2\varrho }}-{\left(\frac{1}{2}\right)}^{s+1}}{s+1}}+{\displaystyle \frac{{\left(\frac{1}{2}\right)}^{s+1}-{\left(1-{\left(\frac{1}{3}\right)}^{\frac{1}{2-2\varrho }}\right)}^{s+1}}{s+1}}\right),\hfill & \mathrm{if}& 0<\varrho \le {\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\\ {\displaystyle \frac{1-2^{2\varrho -s-3}}{s+3-2\varrho }}+\mathbf{B}(3-2\varrho ,s+1)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right)-{\displaystyle \frac{1}{3(s+1)}}\hfill & \mathrm{if}& \varrho >{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

(9)

where $\mathbf{B}(·,·)$ and ${\mathbf{B}}_{\kappa }(·,·)$ represent the Beta and incomplete Beta functions, expressed as

$$\mathbf{B}({\mathfrak{c}}_1,{\mathfrak{c}}_2)=\underset{0}{\stackrel{1}{\int }}{\mu }^{{\mathfrak{c}}_1-1}{(1-\mu )}^{{\mathfrak{c}}_2-1}\mathrm{d}\mu$$

(10)

and

$${\mathbf{B}}_{\kappa }({\mathfrak{c}}_1,{\mathfrak{c}}_2)=\underset{0}{\stackrel{\kappa }{\int }}{\mu }^{{\mathfrak{c}}_1-1}{(1-\mu )}^{{\mathfrak{c}}_2-1}\mathrm{d}\mu ,\mathrm{for}0<\kappa <1,$$

respectively.

Thus, the proof is completed. □

Corollary 1.

If we set $s=1$, Theorem 2 becomes

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{5{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }}{24}}\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right)+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\mathcal{D}}_1\left(\varrho ,1\right)+{\mathcal{D}}_2\left(\varrho ,1\right)\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where ${\mathcal{D}}_1(\varrho ,1)$ and ${\mathcal{D}}_2(\varrho ,1)$ are defined as in (8) and (9), respectively.

Corollary 2.

If we attempt to tend $\varrho \to 1^{-}$, Theorem 2 gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{\left(s+1\right)\left(s+2\right)}}\left({\displaystyle \frac{2s+1}{3}}+{\left({\displaystyle \frac{1}{2}}\right)}^{s+1}\right)\left(\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|\right).\hfill \end{array}$$

Remark 2.

For $s=1$, Corollary 2 yields the same result presented in [7] (Corollary 2.4).

Corollary 3.

By tending $\varrho \to 0^{+}$, Theorem 2 becomes

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{2}}\mathcal{G}\left(s\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathcal{G}\left(s\right)=& {\displaystyle \frac{4s^2+14s+6}{3\left(s+1\right)\left(s+2\right)\left(s+3\right)}}+{\displaystyle \frac{2}{3\left(s+1\right)}}{\left({\displaystyle \frac{1}{3}}\right)}^{\frac{s+1}{2}}-{\displaystyle \frac{2}{\left(s+3\right)}}{\left({\displaystyle \frac{1}{3}}\right)}^{\frac{s+3}{2}}\hfill \\ \hfill & +{\displaystyle \frac{4}{\left(s+1\right)\left(s+2\right)}}{\left({\displaystyle \frac{1}{3}}\right)}^{\frac{1}{2}}{\left({\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}}}\right)}^{s+2}+{\displaystyle \frac{4}{\left(s+1\right)\left(s+2\right)\left(s+3\right)}}{\left({\displaystyle \frac{\sqrt{3}-1}{\sqrt{3}}}\right)}^{s+3}.\hfill \end{array}$$

Corollary 4.

In Corollary 3, if we take $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{108\sqrt{3}}}\left(9\sqrt{3}+24\right)\left(\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|+\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|\right).\hfill \end{array}$$

Theorem 3.

Let $\mathcal{F}:\mathbf{I}\subset {\mathbb{R}}^{+}\to \mathbb{R}$ be a twice differentiable function on ${\mathbf{I}}^{\circ }$, where ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in {\mathbf{I}}^{\circ }$ satisfying ${\mathfrak{d}}_1<{\mathfrak{d}}_2$, and let $\mathcal{F},{\mathcal{F}}^{\prime },{\mathcal{F}}^{″}\in L_1\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$. If ${\left|{\mathcal{F}}^{\prime }\right|}^q$ and ${\left|{\mathcal{F}}^{″}\right|}^q$ are s-convex on $\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$ for $q>1$ with ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{{\left(2/3\right)}^{p+1}-{\left(1/6\right)}^{p+1}}{p+1}}\right)}^{\frac{1}{p}}\left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Proof. 

Applying Hölder’s inequality to the identity presented in Lemma 1, then using the s-convexity of ${\left|{\mathcal{F}}^{\prime }\right|}^q$ and ${\left|{\mathcal{F}}^{″}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|\mu -{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}+{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{{\displaystyle \frac{1}{q}}}\right.+{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left[{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}+\left.{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\right)\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|\mu -{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\mu }^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right)\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left({\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\left[{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.\left.{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left({\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left(1-\mu \right)}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{{\left(2/3\right)}^{p+1}-{\left(1/6\right)}^{p+1}}{p+1}}\right)}^{\frac{1}{p}}\left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}+\left.{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}\right]\right).\hfill \end{array}$$

The proof is completed. □

Corollary 5.

By choosing $s=1$, Theorem 3 yields

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{{\left(2/3\right)}^{p+1}-{\left(1/6\right)}^{p+1}}{p+1}}\right)}^{\frac{1}{p}}\left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+3{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{3{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\right)\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+3{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{3{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Corollary 6.

If we attempt to tend $\varrho \to 1^{-}$, Theorem 3 gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\hfill \\ \hfill \le & \left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right){\left({\displaystyle \frac{{\left(2/3\right)}^{p+1}-{\left(1/6\right)}^{p+1}}{p+1}}\right)}^{{\displaystyle \frac{1}{p}}}\left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{{\displaystyle \frac{1}{q}}}+{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{{\displaystyle \frac{1}{q}}}\right].\hfill \end{array}$$

Moreover, by setting $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\hfill \\ \hfill \le & \left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right){\left({\displaystyle \frac{{\left(2/3\right)}^{p+1}-{\left(1/6\right)}^{p+1}}{p+1}}\right)}^{\frac{1}{p}}\left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+3{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{3{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}\right],\hfill \end{array}$$

which is equivalent to the result presented in the second inequality of Corollary 2.8 from [7].

Corollary 7.

If we attempt to tend $\varrho \to 0^{+}$, Theorem 3 gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^2}{4}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left({\displaystyle \frac{2}{3}}-{\mu }^2\right)}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^2-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\right]\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{\left(2^{s+1}-1\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{2^{s+1}(s+1)}}\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Moreover, by choosing $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^2}{4}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}{\left({\displaystyle \frac{2}{3}}-{\mu }^2\right)}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}{\left|{\mu }^2-{\displaystyle \frac{1}{3}}\right|}^p\mathrm{d}\mu \right)}^{\frac{1}{p}}\right]\hfill \\ \hfill & \times \left[{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+3{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{3{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q}{8}}\right)}^{\frac{1}{q}}\right].\hfill \end{array}$$

Theorem 4.

Let $\mathcal{F}:\mathbf{I}\subset {\mathbf{R}}^{+}\to \mathbb{R}$ be a twice differentiable function on ${\mathbf{I}}^{\circ }$, where ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in {\mathbf{I}}^{\circ }$ satisfying ${\mathfrak{d}}_1<{\mathfrak{d}}_2$, and let $\mathcal{F},{\mathcal{F}}^{\prime },{\mathcal{F}}^{″}\in L_1\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$. If ${\left|{\mathcal{F}}^{\prime }\right|}^q$ and ${\left|{\mathcal{F}}^{″}\right|}^q$ are s-convex on $\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$ for $q>1$, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{5}{24}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{(1-\varrho ){({\mathfrak{d}}_2-{\mathfrak{d}}_1)}^{2-\varrho }}{4}}\left({\left({\displaystyle \frac{{\mathcal{D}}_1(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_3(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_4(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & \left.+{\left({\mathcal{D}}_4(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_3(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left({\displaystyle \frac{{\mathcal{D}}_2(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_5(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_6(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\left.+{\left({\mathcal{D}}_6(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_5(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\right).\hfill \end{array}$$

Here, ${\mathcal{D}}_1$ and ${\mathcal{D}}_2$ are defined as in (8) and (9), and ${\mathcal{C}}_1\left(s\right)$, ${\mathcal{C}}_2\left(s\right)$, ${\mathcal{D}}_3$, ${\mathcal{D}}_4$, ${\mathcal{D}}_5$ and ${\mathcal{D}}_6$ are given by

$${\mathcal{C}}_1\left(s\right)=\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|{(1-\mu )}^s={\displaystyle \frac{1-s+(2s+1)2^{s+2}}{3\times 2^{s+2}(s+1)(s+2)}},$$

(11)

$${\mathcal{C}}_2\left(s\right)=\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|{\mu }^s={\displaystyle \frac{s+5}{3\times 2^{s+2}(s+1)(s+2)}},$$

(12)

$$\begin{array}{cc}\hfill {\mathcal{D}}_3(\varrho ,s)=& \underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|{(1-\mu )}^s\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \frac{2-2^{-s}}{3(s+1)}}-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right),\hfill & if& \varrho \ge {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\\ -2\left[{\displaystyle \frac{2}{3}}{\displaystyle \frac{1-{\left(1-{\left(\frac{2}{3}\right)}^{\frac{1}{2-2\varrho }}\right)}^{s+1}}{s+1}}-{\mathbf{B}}_{{\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,s+1\right)\right],\hfill & if& 0<\varrho <{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

(13)

$${\mathcal{D}}_4(\varrho ,s)={\mathcal{D}}_1(\varrho ,s)-{\mathcal{D}}_3(\varrho ,s),$$

(14)

$$\begin{array}{cc}\hfill {\mathcal{D}}_5(\varrho ,s)=& \underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|{(1-\mu )}^s\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \mathbf{B}\left(3-2\varrho ,s+1\right)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right)-\frac{2^{-(s+1)}}{3(s+1)},}\hfill & if& \varrho >{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\\ {\displaystyle -2[{\mathbf{B}}_{{\left(\frac{1}{3}\right)}^{\frac{1}{2-2\varrho }}}\left(3-2\varrho ,s+1\right)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,s+1\right)}\hfill & & \\ {\displaystyle -\frac{2^{-(s+1)}-{\left(1-{\left(\frac{1}{3}\right)}^{\frac{1}{2-2\varrho }}\right)}^{s+1}}{3(s+1)}],}\hfill & if& 0<\varrho \le {\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

(15)

and

$${\mathcal{D}}_6(\varrho ,s)={\mathcal{D}}_2(\varrho ,s)-{\mathcal{D}}_5(\varrho ,s).$$

(16)

Proof. 

Applying the power mean inequality to the identity presented in Lemma 1, then using the s-convexity of ${\left|{\mathcal{F}}^{\prime }\right|}^q$ and ${\left|{\mathcal{F}}^{″}\right|}^q$, we obtain

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|{\left|{\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|{\left|{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|{\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & \left.+{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|{\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|{\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left.\left.+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|{\left|{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|}^q\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\right)\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left({(1-\mu )}^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & +\left.{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left({\mu }^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{(1-\mu )}^s{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left({\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left({(1-\mu )}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & \left.+{\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left({\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{(1-\mu )}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\mathrm{d}\mu \right)}^{1-\frac{1}{q}}\left[{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left({(1-\mu )}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & \left.\left.+{\left(\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left({\mu }^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{(1-\mu )}^s{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)\mathrm{d}\mu \right)}^{\frac{1}{q}}\right]\right)\hfill \\ \hfill =& {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{5}{24}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{(1-\varrho ){({\mathfrak{d}}_2-{\mathfrak{d}}_1)}^{2-\varrho }}{4}}\left({\left({\displaystyle \frac{{\mathcal{D}}_1(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_3(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_4(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & \left.+{\left({\mathcal{D}}_4(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_3(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left({\displaystyle \frac{{\mathcal{D}}_2(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_5(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_6(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\left.+{\left({\mathcal{D}}_6(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_5(\varrho ,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\right),\hfill \end{array}$$

where we have used (8)–(16). The proof is completed. □

Corollary 8.

By choosing $s=1$, Theorem 4 yields

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }{\left({\displaystyle \frac{5}{24}}\right)}^{1-\frac{1}{q}}\left[{\left({\displaystyle \frac{1}{6}}{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\displaystyle \frac{1}{24}}{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{1}{24}}{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\displaystyle \frac{1}{6}}{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\displaystyle \frac{(1-\varrho ){({\mathfrak{d}}_2-{\mathfrak{d}}_1)}^{2-\varrho }}{4}}\left({\left({\displaystyle \frac{{\mathcal{D}}_1(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_3(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_4(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\right.\hfill \\ \hfill & \left.+{\left({\mathcal{D}}_4(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_3(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left({\displaystyle \frac{{\mathcal{D}}_2(\varrho ,0)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_5(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_6(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\left.+{\left({\mathcal{D}}_6(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_5(\varrho ,1){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{D}}_1\left(\varrho ,1\right)=& \underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \frac{1}{3}}-{\displaystyle \frac{2^{2\varrho -4}}{4-2\varrho }}-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,2\right)\hfill & if& 0<\varrho \le {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\\ -2\left[{\displaystyle \frac{1}{3}}\left({\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{2}{2-2\varrho }}}+1-{\left(1-{\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}\right)}^2\right)-\left({\displaystyle \frac{{\left(\frac{2}{3}\right)}^{\frac{4-2\varrho }{2-2\varrho }}}{4-2\varrho }}+{\mathbf{B}}_{{\left(\frac{2}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,2\right)\right)\right],\hfill & if& \varrho >{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}}\end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{D}}_2\left(\varrho ,1\right)=& \underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\displaystyle \frac{1}{3}}-{\mu }^{2-2\varrho }\right|\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \frac{1-2^{2\varrho -4}}{4-2\varrho }}+\mathbf{B}(3-2\varrho ,2)-{\displaystyle \frac{1}{6}}\hfill & & \\ -2\left[{\displaystyle \frac{{\left(\frac{1}{3}\right)}^{\frac{4-2\varrho }{2-2\varrho }}-{\left(\frac{1}{2}\right)}^{4-2\varrho }}{4-2\varrho }}+{\mathbf{B}}_{{\left(\frac{1}{3}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,2\right)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,2\right)\right]\hfill & & \\ +{\displaystyle \frac{1}{3}}\left({\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{2}{2-2\varrho }}}-{\left(1-{\left({\displaystyle \frac{1}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}\right)}^2\right),\hfill & if& 0<\varrho \le {\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\\ {\displaystyle \frac{1-2^{2\varrho -4}}{4-2\varrho }}+\mathbf{B}(3-2\varrho ,2)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,2\right)-{\displaystyle \frac{1}{6}}\hfill & if& \varrho >{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathcal{D}}_3(\varrho ,1)=& \underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|(1-\mu )\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{ccc}{\displaystyle \frac{1}{4}}-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,2\right),\hfill & if& \varrho \ge {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\\ -2\left[{\displaystyle \frac{1}{3}}\left(1-{\left(1-{\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}\right)}^2\right)-{\mathbf{B}}_{{\left({\displaystyle \frac{2}{3}}\right)}^{{\displaystyle \frac{1}{2-2\varrho }}}}\left(3-2\varrho ,2\right)\right],\hfill & if& 0<\varrho <{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

$${\mathcal{D}}_4(\varrho ,1)={\mathcal{D}}_1(\varrho ,1)-{\mathcal{D}}_3(\varrho ,1),$$

$$\begin{array}{cc}\hfill {\mathcal{D}}_5(\varrho ,1)=& \underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|(1-\mu )\mathrm{d}\mu \hfill \\ & =\left\{\begin{array}{ccc}{\displaystyle \mathbf{B}\left(3-2\varrho ,2\right)-{\mathbf{B}}_{\frac{1}{2}}\left(3-2\varrho ,2\right)-\frac{1}{24},}\hfill & if& \varrho >{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\\ {\displaystyle -2\left[{\mathbf{B}}_{{\left(\frac{1}{3}\right)}^{\frac{1}{2-2\varrho }}}\left(3-2\varrho ,2\right)-\frac{\frac{1}{4}-{\left(1-{\left(\frac{1}{3}\right)}^{\frac{1}{2-2\varrho }}\right)}^2}{6}\right],}\hfill & if& 0<\varrho \le {\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\end{array}\right.\hfill \end{array}$$

$${\mathcal{D}}_6(\varrho ,1)={\mathcal{D}}_2(\varrho ,1)-{\mathcal{D}}_5(\varrho ,1).$$

Corollary 9.

If we attempt to tend $\varrho \to 1^{-}$, Theorem 4 gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\hfill \\ \hfill \le & ({\mathfrak{d}}_2-{\mathfrak{d}}_1){\left({\displaystyle \frac{5}{24}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathcal{C}}_2\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{C}}_1\left(s\right){\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right],\hfill \end{array}$$

where ${\mathcal{C}}_1\left(s\right)$ and ${\mathcal{C}}_2\left(s\right)$ are defined as in (11) and (12).

Moreover, by setting $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\hfill \\ \hfill \le & ({\mathfrak{d}}_2-{\mathfrak{d}}_1){\left({\displaystyle \frac{5}{24}}\right)}^{1-\frac{1}{q}}\left[{\left({\displaystyle \frac{4{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{24}}\right)}^{\frac{1}{q}}+{\left({\displaystyle \frac{{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)\right|}^q+4{\left|{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right|}^q}{24}}\right)}^{\frac{1}{q}}\right],\hfill \end{array}$$

which is equivalent to the result presented in the second inequality of Corollary 2.11 from [7].

Corollary 10.

If we attempt to tend $\varrho \to 0^{+}$, Theorem 4 gives

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{({\mathfrak{d}}_2-{\mathfrak{d}}_1)}^2}{4}}\left({\left({\displaystyle \frac{{\mathcal{D}}_1(0,s)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_3(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_4(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.+{\left({\mathcal{D}}_4(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_3(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left({\displaystyle \frac{{\mathcal{D}}_2(0,s)}{2}}\right)}^{1-\frac{1}{q}}\left[{\left({\mathcal{D}}_5(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_6(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\left.+{\left({\mathcal{D}}_6(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\mathcal{D}}_5(0,s){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\right),\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{D}}_1(0,s)=& {\displaystyle \frac{2}{3(s+1)}}-{\displaystyle \frac{2^{-(s+3)}}{s+3}}-\mathbf{B}\frac{1}{2}(3,s+1),\hfill \\ \hfill {\mathcal{D}}_2(0,s)=& {\displaystyle \frac{1-2^{-(s+3)}}{s+3}}+\mathbf{B}(3,s+1)-{\displaystyle \frac{1}{3(s+1)}}-2\left[{\displaystyle \frac{{\left(\frac{1}{\sqrt{3}}\right)}^{s+3}-2^{-(s+3)}}{s+3}}+\mathbf{B}\frac{1}{\sqrt{3}}(3,s+1)-\mathbf{B}\frac{1}{2}(3,s+1)\right]\hfill \\ \hfill & +{\displaystyle \frac{2}{3(s+1)}}\left({\left({\displaystyle \frac{1}{\sqrt{3}}}\right)}^{s+1}-{\left(1-{\displaystyle \frac{1}{\sqrt{3}}}\right)}^{s+1}\right),\hfill \\ \hfill {\mathcal{D}}_3(0,s)=& {\displaystyle \frac{2-2^{-s}}{3(s+1)}}-\mathbf{B}\frac{1}{2}(3,s+1),\hfill \\ \hfill {\mathcal{D}}_4(0,s)=& {\displaystyle \frac{2^{-s}}{3(s+1)}}-{\displaystyle \frac{2^{-(s+3)}}{s+3}},\hfill \\ \hfill {\mathcal{D}}_5(0,s)=& \mathbf{B}(3,s+1)-\mathbf{B}\frac{1}{2}(3,s+1)-{\displaystyle \frac{2^{-(s+1)}}{3(s+1)}}-2\left[\mathbf{B}\frac{1}{\sqrt{3}}(3,s+1)-{\mathbf{B}}_{\frac{1}{2}}(3,s+1)-{\displaystyle \frac{2^{-(s+1)}-{\left(1-\frac{1}{\sqrt{3}}\right)}^{s+1}}{3(s+1)}}\right],\hfill \\ \hfill {\mathcal{D}}_6(0,s)=& {\displaystyle \frac{1+2^{-(s+3)}-2{\left(\frac{1}{\sqrt{3}}\right)}^{s+3}}{s+3}}+{\displaystyle \frac{2{\left(\frac{1}{\sqrt{3}}\right)}^{s+1}-2^{-(s+1)}-1}{3(s+1)}}.\hfill \end{array}$$

Moreover, by choosing $s=1$, we get

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\hfill \\ \hfill \le & {\displaystyle \frac{{({\mathfrak{d}}_2-{\mathfrak{d}}_1)}^2}{4}}\left({\left({\displaystyle \frac{7}{48}}\right)}^{1-\frac{1}{q}}\left[{\left({\displaystyle \frac{43}{192}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\displaystyle \frac{13}{192}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.+{\left({\displaystyle \frac{13}{192}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\displaystyle \frac{43}{192}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\hfill \\ \hfill & +{\left({\displaystyle \frac{32\sqrt{3}-27}{432}}\right)}^{1-\frac{1}{q}}\left[{\left(\left({\displaystyle \frac{4\sqrt{3}}{27}}-{\displaystyle \frac{137}{576}}\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+{\displaystyle \frac{65}{576}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right.\hfill \\ \hfill & \left.\left.+{\left({\displaystyle \frac{65}{576}}{\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_1\right)\right|}^q+\left({\displaystyle \frac{4\sqrt{3}}{27}}-{\displaystyle \frac{137}{576}}\right){\left|{\mathcal{F}}^{″}\left({\mathfrak{d}}_2\right)\right|}^q\right)}^{\frac{1}{q}}\right]\right).\hfill \end{array}$$

The following theorem presents a proportional Caputo-hybrid Milne-type inequality for Lipschitzian functions. We recall that a function $\mathcal{F}:[{\mathfrak{d}}_1,{\mathfrak{d}}_2]\to \mathbb{R}$ is said to be $\mathrm{L}$-Lipschitz continuous on the interval $[{\mathfrak{d}}_1,{\mathfrak{d}}_2]$ if there exists a real constant $\mathrm{L}\ge 0$ such that

$$\left|\mathcal{F}\left(\mathfrak{v}\right)-\mathcal{F}\left(\mathfrak{u}\right)\right|\le \mathrm{L}|\mathfrak{v}-\mathfrak{u}|,\mathrm{for}\mathrm{all}\mathfrak{v},\mathfrak{u}\in [{\mathfrak{d}}_1,{\mathfrak{d}}_2].$$

Theorem 5.

Let $\mathcal{F}:\mathbf{I}\subset {\mathbf{R}}^{+}\to \mathbb{R}$ be a twice differentiable function on ${\mathbf{I}}^{\circ }$, where ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in {\mathbf{I}}^{\circ }$ satisfying ${\mathfrak{d}}_1<{\mathfrak{d}}_2$, and let $\mathcal{F},{\mathcal{F}}^{\prime },{\mathcal{F}}^{″}\in L_1\left[{\mathfrak{d}}_1,{\mathfrak{d}}_2\right]$. If ${\mathcal{F}}^{\prime }$ and ${\mathcal{F}}^{″}$ are $\mathrm{L}$- and $\mathrm{H}$-Lipschitzian functions, respectively, then the following inequality holds:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\displaystyle \frac{{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }}{8}}\mathrm{L}+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{3-\varrho }}{4}}\mathcal{E}\left(\varrho \right)\mathrm{H},\hfill \end{array}$$

where $\mathcal{E}\left(\varrho \right)$ is expressed as in (17).

Proof. 

From Lemma 1, and the fact that ${\mathcal{F}}^{\prime }$ and ${\mathcal{F}}^{″}$ are $\mathrm{L}$-Lipschitzian and $\mathrm{H}$-Lipschitzian functions, respectively, we have the next absolute values’ inequality:

$$\begin{array}{cc}\hfill & \left|{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,\mathcal{F}\right)+{\displaystyle \frac{1-\varrho }{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho -1}}}\mathcal{M}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2,{\mathcal{F}}^{\prime }\right)-{\displaystyle \frac{\Gamma \left(1-\varrho \right)}{2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1-\varrho }}}\left({\mathcal{D}}_{{\mathfrak{d}}_2^{-}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_1\right)+{\mathcal{D}}_{{\mathfrak{d}}_1^{+}}^{\varrho }\mathcal{F}\left({\mathfrak{d}}_2\right)\right)\right|\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{\varrho }\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left|{\mathcal{F}}^{\prime }\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{\prime }\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\mathrm{d}\mu \hfill \\ \hfill & +{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{2-\varrho }}{4}}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\mathrm{d}\mu \right.\hfill \\ \hfill & \left.+\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left|{\mathcal{F}}^{″}\left(\left(1-\mu \right){\mathfrak{d}}_1+\mu {\mathfrak{d}}_2\right)-{\mathcal{F}}^{″}\left(\mu {\mathfrak{d}}_1+\left(1-\mu \right){\mathfrak{d}}_2\right)\right|\mathrm{d}\mu \right)\hfill \\ \hfill \le & {\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }\mathrm{L}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|\left|2\mu -1\right|\mathrm{d}\mu \right)+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{3-\varrho }}{4}}\mathrm{H}\left(\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left|2\mu -1\right|\mathrm{d}\mu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left|2\mu -1\right|\mathrm{d}\mu \right)\hfill \\ \hfill =& {\displaystyle \frac{{\varrho }^2{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }}{8}}\mathrm{L}+{\displaystyle \frac{\left(1-\varrho \right){\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{3-\varrho }}{4}}\mathcal{E}\left(\varrho \right)\mathrm{H},\hfill \end{array}$$

where we have used

$$\underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|\mu -{\displaystyle \frac{2}{3}}\right|·\left|1-2\mu \right|\mathrm{d}\mu ={\displaystyle \frac{1}{8}},$$

and

$$\begin{array}{cc}\hfill \mathcal{E}\left(\varrho \right)=& \underset{0}{\stackrel{\frac{1}{2}}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{2}{3}}\right|\left|2\mu -1\right|\mathrm{d}\mu +\underset{\frac{1}{2}}{\stackrel{1}{\int }}\left|{\mu }^{2-2\varrho }-{\displaystyle \frac{1}{3}}\right|\left|2\mu -1\right|\mathrm{d}\mu \hfill \\ \hfill =& \left\{\begin{array}{cc}{\displaystyle \frac{1}{12}+\frac{1}{2-\varrho }-\frac{1}{3-2\varrho },}\hfill & \mathrm{if}{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}}\le \varrho \le {\displaystyle \frac{\ln 8-\ln 3}{\ln 4}},\hfill \\ {\displaystyle \frac{1}{12}+\frac{1}{2-\varrho }-\frac{1}{3-2\varrho }-2\left[\frac{2}{3}\left({\mu }_0-{\mu }_0^2\right)-\frac{{\mu }_0^{3-2\varrho }}{3-2\varrho }+\frac{2{\mu }_0^{4-2\varrho }}{4-2\varrho }\right],}\hfill & \mathrm{if}{\displaystyle \frac{\ln 8-\ln 3}{\ln 4}}<\varrho \le 1,\hfill \\ {\displaystyle \frac{1}{12}+\frac{1}{2-\varrho }-\frac{1}{3-2\varrho }-2\left[\frac{2(1-{\mu }_1^{4-2\varrho })}{4-2\varrho }-\frac{1-{\mu }_1^{3-2\varrho }}{3-2\varrho }-\frac{{(1-{\mu }_1)}^2}{3}\right],}\hfill & \mathrm{if}0<\varrho <{\displaystyle \frac{\ln 4-\ln 3}{\ln 4}},\hfill \end{array}\right.\hfill \end{array}$$

(17)

where

$${\mu }_0={\left({\displaystyle \frac{2}{3}}\right)}^{1/(2-2\varrho )},\mathrm{and}{\mu }_1={\left({\displaystyle \frac{1}{3}}\right)}^{1/(2-2\varrho )}.$$

The proof is completed. □

Corollary 11.

If we attempt to tend $\varrho \to 1^{-}$, Theorem 5 gives

$$\left|{\displaystyle \frac{1}{3}}\left(2\mathcal{F}\left({\mathfrak{d}}_1\right)-\mathcal{F}\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2\mathcal{F}\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\underset{{\mathfrak{d}}_1}{\stackrel{{\mathfrak{d}}_2}{\int }}\mathcal{F}\left(\varpi \right)\mathrm{d}\varpi \right|\le {\displaystyle \frac{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{1+\varrho }}{8}}\mathrm{L}.$$

Corollary 12.

If we attempt to tend $\varrho \to 0^{+}$, Theorem 3 gives

$$\left|{\displaystyle \frac{1}{3}}\left(2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_1\right)-{\mathcal{F}}^{\prime }\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)+2{\mathcal{F}}^{\prime }\left({\mathfrak{d}}_2\right)\right)-{\displaystyle \frac{\mathcal{F}\left({\mathfrak{d}}_2\right)-\mathcal{F}\left({\mathfrak{d}}_1\right)}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\le {\displaystyle \frac{{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^{3-\varrho }}{4}}\left({\displaystyle \frac{11}{12}}-{\displaystyle \frac{14\sqrt{3}}{27}}\right)\mathrm{H}.$$

4. Example and Applications

We complement the theoretical developments with a concrete numerical example, including graphical visualization, and discuss potential applications in numerical integration and error estimation.

4.1. Illustrative Example

To demonstrate the effectiveness of the derived inequalities, we present a numerical example in which the theoretical bounds are computed explicitly and visualized through graphical plots.

Example 1.

Let $\mathcal{F}:[{\mathfrak{d}}_1,{\mathfrak{d}}_2]\to \mathbb{R}$ be defined on the interval $[{\mathfrak{d}}_1,{\mathfrak{d}}_2]=[0,1]$ by

$$\mathcal{F}\left(\varpi \right)={\displaystyle \frac{{\varpi }^{s+2}}{s+2}},s\in (0,1].$$

This choice satisfies the required hypotheses: the first derivative ${\mathcal{F}}^{\prime }\left(\varpi \right)={\varpi }^{s+1}$ and the second derivative ${\mathcal{F}}^{″}\left(\varpi \right)=(s+1){\varpi }^s$ are both s-convex functions on $[0,1]$.

Consequently, invoking Theorem 2 yields

$$\begin{array}{cc}\hfill & \left|{\displaystyle \frac{{\varrho }^2}{3(s+2)}}\left(2-{\left({\displaystyle \frac{1}{2}}\right)}^{s+2}\right)+{\displaystyle \frac{1-\varrho }{6}}\left(2-{\left({\displaystyle \frac{1}{2}}\right)}^{s+1}\right)-{\displaystyle \frac{{(1-\varrho )}^2}{2}}\left[\mathbf{B}(s+2,2-2\varrho )+{\displaystyle \frac{1}{3-2\varrho +s}}\right]-{\displaystyle \frac{{\varrho }^2}{(s+2)(s+3)}}\right|\hfill \\ \hfill \le & {\varrho }^2\mathcal{C}\left(s\right)+{\displaystyle \frac{(1-\varrho )(s+1)}{4}}\left({\mathcal{D}}_1(\varrho ,s)+{\mathcal{D}}_2(\varrho ,s)\right),\hfill \end{array}$$

(18)

where $\mathcal{C}\left(s\right)$, ${\mathcal{D}}_1(\varrho ,s)$, ${\mathcal{D}}_2(\varrho ,s)$, and $\mathbf{B}(s+2,2-2\rho )$ are defined as in (7)–(9) and (10), respectively.

Figure 1 displays the left- and right-hand sides of inequality (18) over the domain $\varrho \in (0,1)$ and $s\in (0,1]$. As illustrated, the right-hand side consistently dominates the left-hand side, thereby supporting the correctness of the established inequality.

4.2. Applications

We further explore practical implications of our results, focusing on their potential use in error estimation for fractional quadrature formulas and in the analysis of numerical schemes involving proportional Caputo-hybrid operators. We begin by recalling the following special means of arbitrary real numbers ${\mathfrak{d}}_1$ and ${\mathfrak{d}}_2$:

$$\begin{array}{cc}\hfill \mathrm{Arithmetic}\mathrm{mean}:& \mathcal{A}({\mathfrak{d}}_1,{\mathfrak{d}}_2)={\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}},\hfill \\ \hfill \mathrm{Geometric}\mathrm{mean}:& \mathcal{G}({\mathfrak{d}}_1,{\mathfrak{d}}_2)=\sqrt{{\mathfrak{d}}_1{\mathfrak{d}}_2},\hfill \\ \hfill \mathrm{Harmonic}\mathrm{mean}:& \mathcal{H}({\mathfrak{d}}_1,{\mathfrak{d}}_2)={\displaystyle \frac{2{\mathfrak{d}}_1{\mathfrak{d}}_2}{{\mathfrak{d}}_1+{\mathfrak{d}}_2}},\hfill \\ \hfill \mathrm{Logarithmic}\mathrm{mean}:& \mathcal{L}({\mathfrak{d}}_1,{\mathfrak{d}}_2)={\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{\ln {\mathfrak{d}}_2-\ln {\mathfrak{d}}_1}},{\mathfrak{d}}_1,{\mathfrak{d}}_2>0\mathrm{with}{\mathfrak{d}}_1\ne {\mathfrak{d}}_2,\hfill \\ \hfill p\text{-}\mathrm{Logarithmic}\mathrm{mean}:& {\mathcal{L}}_p({\mathfrak{d}}_1,{\mathfrak{d}}_2)={\displaystyle \frac{{\mathfrak{d}}_2^{p+1}-{\mathfrak{d}}_1^{p+1}}{(p+1)({\mathfrak{d}}_2-{\mathfrak{d}}_1)}},{\mathfrak{d}}_1,{\mathfrak{d}}_2>0\mathrm{with}{\mathfrak{d}}_1\ne {\mathfrak{d}}_2\mathrm{and}p\in \mathbb{R}\setminus \{-1,0\}.\hfill \end{array}$$

Proposition 1.

Let ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in \mathbb{R}$ with $0<{\mathfrak{d}}_1<{\mathfrak{d}}_2$, then we have

$$\left|{\displaystyle \frac{4}{3}}\mathcal{A}\left({\displaystyle \frac{1}{{\mathfrak{d}}_1}},{\displaystyle \frac{1}{{\mathfrak{d}}_2}}\right)-{\displaystyle \frac{1}{3{\mathcal{G}}^2\left({\mathfrak{d}}_1,{\mathfrak{d}}_2\right)}}\mathcal{H}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2\right)-{\left({\mathfrak{d}}_2-{\mathfrak{d}}_1\right)}^2{\mathcal{L}}^{-1}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2\right)\right|\le {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{18}}\left(3+8{\left({\displaystyle \frac{1}{3}}\right)}^{\frac{1}{2}}\right){\mathcal{H}}^{-1}\left({\mathfrak{d}}_1^2,{\mathfrak{d}}_2^2\right).$$

Proof. 

Applying Corollary 4 to the function $\mathcal{F}\left(\varpi \right)=\ln \varpi$ yields

$$\left|{\displaystyle \frac{1}{3}}\left({\displaystyle \frac{2}{{\mathfrak{d}}_1}}-{\displaystyle \frac{2}{{\mathfrak{d}}_1+{\mathfrak{d}}_2}}+{\displaystyle \frac{2}{{\mathfrak{d}}_2}}\right)-{\displaystyle \frac{\ln {\mathfrak{d}}_2-\ln {\mathfrak{d}}_1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}\right|\le {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{108\sqrt{3}}}\left(9\sqrt{3}+24\right)\left({\displaystyle \frac{1}{{\mathfrak{d}}_1^2}}+{\displaystyle \frac{1}{{\mathfrak{d}}_2^2}}\right).$$

The assertion is obtained using the facts that

$${\displaystyle \frac{1}{2}}\left({\displaystyle \frac{1}{{\mathfrak{d}}_1}}+{\displaystyle \frac{1}{{\mathfrak{d}}_2}}\right)=\mathcal{A}\left({\displaystyle \frac{1}{{\mathfrak{d}}_1}},{\displaystyle \frac{1}{{\mathfrak{d}}_2}}\right),$$

$${\displaystyle \frac{2}{{\mathfrak{d}}_1+{\mathfrak{d}}_2}}={\displaystyle \frac{\mathcal{H}({\mathfrak{d}}_1,{\mathfrak{d}}_2)}{{\mathcal{G}}^2({\mathfrak{d}}_1,{\mathfrak{d}}_2)}},$$

$${\displaystyle \frac{\ln {\mathfrak{d}}_2-\ln {\mathfrak{d}}_1}{{\mathfrak{d}}_2-{\mathfrak{d}}_1}}={\mathcal{L}}^{-1}({\mathfrak{d}}_1,{\mathfrak{d}}_2),$$

and

$${\displaystyle \frac{1}{{\mathfrak{d}}_1^2}}+{\displaystyle \frac{1}{{\mathfrak{d}}_2^2}}=2{\mathcal{H}}^{-1}({\mathfrak{d}}_1,{\mathfrak{d}}_2).$$

□

Proposition 2.

Let ${\mathfrak{d}}_1,{\mathfrak{d}}_2\in \mathbb{R}$ with $0<{\mathfrak{d}}_1<{\mathfrak{d}}_2$ and $s\in (0,1]$, then we have

$$\left|4\mathcal{A}\left({\mathfrak{d}}_1^{s+1},{\mathfrak{d}}_2^{s+1}\right)-{\mathcal{A}}^{s+1}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2\right)-3{\mathcal{L}}_{s+1}^{s+1}\left({\mathfrak{d}}_1,{\mathfrak{d}}_2\right)\right|\le {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{s+2}}\left(2s+1+3{\left({\displaystyle \frac{1}{2}}\right)}^{s+1}\right)\left({\mathfrak{d}}_1^s+{\mathfrak{d}}_2^s\right).$$

Proof. 

Applying Corollary 2 to the function $\mathcal{F}\left(\varpi \right)={\varpi }^{s+1}$ yields

$$\left|{\displaystyle \frac{1}{3}}\left(2{\mathfrak{d}}_1^{s+1}-{\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)}^{s+1}+2{\mathfrak{d}}_2^{s+1}\right)-{\displaystyle \frac{{\mathfrak{d}}_2^{s+2}-{\mathfrak{d}}_1^{s+2}}{(s+2)({\mathfrak{d}}_2-{\mathfrak{d}}_1)}}\right|\le {\displaystyle \frac{{\mathfrak{d}}_2-{\mathfrak{d}}_1}{s+2}}\left({\displaystyle \frac{2s+1}{3}}+2^{-(s+1)}\right)\left({\mathfrak{d}}_1^s+{\mathfrak{d}}_2^s\right).$$

The assertion is obtained by multiplying both sides of the above inequality by 3, and using the following equalities:

$${\displaystyle \frac{{\mathfrak{d}}_1^{s+1}+{\mathfrak{d}}_2^{s+1}}{2}}=\mathcal{A}\left({\mathfrak{d}}_1^{s+1},{\mathfrak{d}}_2^{s+1}\right),$$

$${\left({\displaystyle \frac{{\mathfrak{d}}_1+{\mathfrak{d}}_2}{2}}\right)}^{s+1}={\mathcal{A}}^{s+1}({\mathfrak{d}}_1,{\mathfrak{d}}_2),$$

and

$${\displaystyle \frac{{\mathfrak{d}}_2^{s+2}-{\mathfrak{d}}_1^{s+2}}{(s+2)({\mathfrak{d}}_2-{\mathfrak{d}}_1)}}={\mathcal{L}}_{s+1}^{s+1}({\mathfrak{d}}_1,{\mathfrak{d}}_2).$$

□

5. Conclusions

In this study, we have established Milne-type inequalities from the perspective of proportional Caputo-hybrid fractional operators. We treat the first- and second-order derivatives of the function as either convex or Lipschitz continuous. The inequalities derived from the regularity, be it geometric (convexity) or metric (Lipschitz), explain the error structure in fractional quadrature rules and also generalize classical results. The accompanying numerical experiment validates the sharpness of the theoretical bounds, and the suggested applications provide promising pathways in adaptive numerical integration and uncertainty quantification. This work contributes both methodologically and conceptually to the evolving interface between fractional calculus, functional inequalities, and computational mathematics.