Some Refinements of Hermite–Hadamard Type Integral Inequalities Involving Refined Convex Function of the Raina Type

Abstract

The aim of this work is to elaborate and define the idea of refined convex function of the Raina type. In addition, we have attained some associated properties in the manner of the newly introduced idea. To add some more comprehension into the newly investigated definition, we obtain the estimations of the Hermite-Hadamard inequality. For the reader’s interest, we add some remarks regarding the Mittag-Leffer function. During the last four decades, the term Mitag-Leffler function has acquired popularity on account of its many importance in the fields of engineering and science, i.e statistical distribution theory, rheology, electric networks, fluid flow, and probability. The amazing perception regarding this function provides the solution of certain boundary value problems. The asymptotic status of this function plays a very vital performance in various problems of physics associated with fractional calculus. The methodology and amazing tools of this work may serve as an impetus for further research activities in this direction as well.

1. Introduction

For the most part, integral inequalities structure a solid and flourishing field of study inside the enormous field of mathematics. They have taken an interest in the investigation of numerous fields, e.g., decision making in structural engineering, probabilistic problems, and fatigue life. The Hermite-Hadamard inequality guarantees the integrability of convex functions and presents approximations of the mean value of a convex function as well. Moreover, it involves extraordinary consideration, and one needs to see that a portion of the traditional inequalities for means can be acquired from Hadamard’s inequality under the convenience of convex function. Applications of the classical inequalities are ordinary differential equations, integral equations, and partial differential equations, probability theory, etc.

Integral inequality is a fascinating numerical model because of its wide and critical applications in numerical analysis. Furthermore, the amazing perception on the term theory of inequalities always provides proliferating concept and meaningful importance in every branch of applied and pure sciences, for example, numerical analysis, impulsive diffusion equations, coding theory, geometric function theory, and fractional calculus. For the reader’s attention, see the references [1,2,3,4,5,6,7,8,9,10,11,12].

Inspired by the proceeding research work, the aim of this work is to define and explore a new family of convex function called the Refined convex function of the Raina type. We explore and investigate some nice associated properties and refinements of the H-H inequality in the manner of the newly proposed approach.

2. Preliminaries

For the sake of completeness, quality, and readers’ interest, it will be better to examine and elaborate on several definitions, theorems, and remarks in the preliminary section. The aim of this part is to discuss and study some known concepts and definitions, which we need in our investigation in further sections. We start by introducing convex function, Hermite-Hadamard inequality, Raina function, generalized convex function and s-type convex function.

Definition 1

(see [2]).Let $\mathtt{G}:I\to \mathbb{R}$ be a real valued function. A function $\mathtt{G}$ is said to be convex, if

$$\mathtt{G}\left(\lambda {\mathsf{w}}_{\mathsf{1}}+\left(1-\lambda \right){\mathsf{w}}_{\mathsf{2}}\right)\le \lambda \mathtt{G}\left({\mathsf{w}}_{\mathsf{1}}\right)+\left(1-\lambda \right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right),$$

(1)

holds for all ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I$ and $\lambda \in [0,1].$

The most important inequality involving convex function is the Hermite–Hadamard inequality [13] stated as:

Theorem 1.

If $\mathtt{G}:[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]\to \mathbb{R}$ is a convex function, then

$$\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}+{\mathsf{w}}_{\mathsf{2}}}{2}\right)\le \frac{1}{{\mathsf{w}}_{\mathsf{2}}-{\mathsf{w}}_{\mathsf{1}}}{\int }_{{\mathsf{w}}_{\mathsf{1}}}^{{\mathsf{w}}_{\mathsf{2}}}\mathtt{G}\left(x\right)dx\le \frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{1}}\right)+\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)}{2}.$$

(2)

The above inequality (2) is held for concave function if inequality (2) is in the sense of reverse order. Since then, researchers have presented a great interest in inequality, and many numerous improvements and refinements have been shown in the literature. Due to its many perceptions and importance, this inequality has prevailed in an area of deep affection of analysis. For the attraction of the readers, see the following published articles [14,15,16,17,18,19].

Famous mathematician Raina [20] in 2005, first time explored and investigated a family of functions, which is defined by

$${\mathcal{F}}_{\rho ,\lambda }^{\sigma }\left(z\right)={\mathcal{F}}_{\rho ,\lambda }^{\sigma \left(0\right),\sigma \left(1\right),\dots }\left(z\right)=\sum _{k=0}^{+\infty }\frac{\sigma \left(k\right)}{\mathsf{\Gamma }(\rho k+\lambda )}{z}^k,$$

(3)

where $\rho ,\lambda >0,\left|z\right|<R$ and $\sigma =\left(\sigma \right(0),\dots ,\sigma (k),\dots )$. Equation (3) is the refinement of the Kummer and Mittag–Leffler function.

If $\alpha ,\beta$ and $\gamma$ are parameters and choosing $\rho =1,\lambda =0$ and $\sigma \left(k\right)=\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k}{{\left(\gamma \right)}_k}$ for $k=0,1,2,\dots ,$ then

$${\left(\alpha \right)}_k=\frac{\mathsf{\Gamma }(\alpha +k)}{\mathsf{\Gamma }\left(\alpha \right)}=\alpha (\alpha +1)\dots (\alpha +k-1),k=0,1,2,\dots ,$$

and $\left|z\right|\le 1$ (with $z\in \mathbb{C}$), then hypergeometric function is

$$\mathcal{F}(\alpha ,\beta ;\gamma ;z)=\sum _{k=0}^{+\infty }\frac{{\left(\alpha \right)}_k{\left(\beta \right)}_k}{k!{\left(\gamma \right)}_k}{z}^k.$$

(4)

Also, if we choose the value of $\sigma =(1,1,\dots )$ with the condition $\rho =\alpha ,\left(Re\right(\alpha )>0),\lambda =1$, then we have

$$E_{\alpha }\left(z\right)=\sum _{k=0}^{+\infty }\frac{z^k}{\mathsf{\Gamma }(1+\alpha k)}.$$

(5)

Equation (5) is known as the classical Mittag-Leffler function, which randomly exists in the proof of integrals and derivatives in the sense of fractional. This was first explored by Mittag-Leffler and Wiman in 1903 and 1905 respectively. Nowadays, fractional calculus and Mittag-Leffler functions have a wide range of applications and research activities in the subject of physics. Many research papers and the idea regarding these functions has become popular and interesting due to their vast applications. For the reader’s attention, see the references [21,22,23].

Cortez investigated the following a new family of set and function in the mode of Raina’s function in [24,25].

Definition 2

(see [25]).A non-empty set X is called generalized convex, if

$${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in X,$$

(6)

for all ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in X$ and $\lambda \in [0,1].$ Where $\rho ,\lambda >0$ and $\sigma =\left(\sigma \right(0),\dots ,\sigma (k),\dots )$.

Definition 3

(see [25]). Let $\mathtt{G}:X\to \mathbb{R}$ be real-valued function, then $\mathtt{G}$ is called generalized convex function, if

$$\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\right)\le \lambda \mathtt{G}\left({\mathsf{w}}_{\mathsf{1}}\right)+(1-\lambda )\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right),$$

(7)

for all ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in X,$${\mathsf{w}}_{\mathsf{1}}<{\mathsf{w}}_{\mathsf{2}}$ and $\lambda \in [0,1]$. Where $\rho ,\lambda >0$ and $\sigma =\left(\sigma \right(0),\dots ,\sigma (k),\dots )$.

Remark 1.

Choosing ${\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})={\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}>0,$ then we obtain Definition 1.

Definition 4

(see [26]). A nonnegative function $\mathtt{G}:\mathbb{A}\to \mathbb{R}$ is said to s–type convex function if $\forall {\mathsf{w}}_{\mathsf{2}},{\mathsf{w}}_{\mathsf{1}}\in \mathbb{A},$$s\in [0,1]$ and $\lambda \in [0,1],$ if

$$\mathtt{G}\left(\lambda {\mathsf{w}}_{\mathsf{2}}+\left(1-\lambda \right){\mathsf{w}}_{\mathsf{1}}\right)\le \left[1-\left(s\right(1-\lambda \left)\right)\right]\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\left[1-s\lambda \right]\mathtt{G}\left({\mathsf{w}}_{\mathsf{1}}\right).$$

(8)

Motivated by the continuing research journey, the construction of this manuscript is marked as follows. First and foremost, in Section 3, we will examine and investigate the recently introduced ideas of refined convex function of Raina type and its associated properties. In Section 4, on the basis of the lemma, we will obtain estimations of the Hermite–Hadamard inequality with the help of the proposed new definition.

3. Refined Convex Function of Raina Type and Its Properties

Due to the theory of convexity’s numerous applications in applied sciences and optimization issues, it has undergone remarkable development during the past few decades. Even while convexity has yielded a variety of conclusions, the majority of problems in the real world are nonconvex in nature. Studying nonconvex functions, which are roughly close to convex functions, is therefore always worthwhile. Convex functions have received acclaim from numerous well-known mathematicians during the twentieth century, including Jensen, Hermite, Holder, and Stolz. An unprecedented amount of research was done throughout the 20th century, yielding significant findings in the fields of convex analysis, geometric functional analysis, and nonlinear programming.

The goal of this part is to define and examine a new class of convex functions, namely refined convex function of Raina type, as well as to explore the properties of this newly proposed definition.

Definition 5.

Let $\sigma =\left(\sigma \right(0),\dots ,\sigma (k),\dots )$, $\rho ,\lambda >0$, $\lambda \in [0,1]$ and $\mathbb{X}\subseteq [0,b^{\ast }],b^{\ast }>0$ be a generalized convex set w.r.t ${\mathcal{F}}_{\rho ,\lambda }^{\sigma }:\mathbb{X}\times \mathbb{X}\to \mathbb{R}$. Then $\mathtt{G}:\mathbb{X}\to \mathbb{R}$ is known as refined convex function of Raina type for fixed $(\alpha ,m)\in (0,1]\times (0,1]$, if

$$\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\le \left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right),$$

(9)

holds for every $s\in [0,1]$ and ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in \mathbb{X}.$

Remark 2.

(i)

Taking $s=\alpha =m=1$ in above Definition 5, then we obtain a Definition of Cortez ([24] Definition 4 and [25] Definition 4) namely generalized convex function of Raina type.

(ii)

Taking $\alpha =m=1$ and ${\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})={\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}$ in Definition 5, then we attain a published definition namely s–type convex function which is first time explored by İşcan et al. [26].

(iii)

Taking $s=\alpha =m=1$ and ${\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})={\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}$ in above Definition 5, then we attain a definition namely convex function which is explored by Niculescu et al. [2].

Theorem 2.

Let $\mathtt{G},\mathtt{H}:\mathbb{A}=[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]\to \mathbb{R}.$ If $\mathtt{G},\mathtt{H}$ be two refined convex functions of Raina type with $(\alpha ,m)$, then the sum of these functions is refined convex function of Raina type with $(\alpha ,m)$.

Proof. 

Let $\mathtt{G},\mathtt{H}$ be refined convex function of Raina type, then for all ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in \mathbb{A},$$s\in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $\lambda \in [0,1],$ we have

$$\begin{array}{cc}\hfill & \left(\mathtt{G}+\mathtt{H}\right)({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & =\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))+\mathtt{H}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & \le \left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\hfill \\ \hfill & +\left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{H}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{H}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)\left[\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{H}\left({\mathsf{w}}_{\mathsf{2}}\right)\right]+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }\left[m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)+m\mathtt{H}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right]\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)(\mathtt{G}+\mathtt{H})\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m(\mathtt{G}+\mathtt{H})\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right).\hfill \end{array}$$

This completes the proof. □

Theorem 3.

Scalar multiplication of refined convex function of Raina type is again refined convex function of Raina type.

Proof. 

Let $\mathtt{G}$ be refined convex function of Raina type, then for all ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in \mathbb{A},$$s\in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$, $c\in \mathbb{R}$, $(c\ge 0)$, and $\lambda \in [0,1],$ we have

$$\begin{array}{cc}\hfill & \left(c\mathtt{G}\right)({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & \le c\left[\left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right]\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)c\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }cm\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)\left(c\mathtt{G}\right)\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\left(c\mathtt{G}\right)\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right).\hfill \end{array}$$

This completes the required proof. □

Theorem 4.

Assume that $\mathtt{G}:\mathbb{A}\to \mathbb{Y}$ be an refined convex function of Raina type and $\mathtt{H}:\mathbb{Y}\to \mathbb{R}$ is an increasing function. Then $\mathtt{H}\circ \mathtt{G}$ is refined convex function of Raina type for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$.

Proof. 

$$\begin{array}{cc}\hfill & \left(\mathtt{H}\circ \mathtt{G}\right)({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & =\mathtt{H}(\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})))\hfill \\ & \le \mathtt{H}\left[\left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right]\hfill \\ & \le \left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{H}\left(\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{H}\left(m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right)\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)(\mathtt{H}\circ \mathtt{G})\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }{m}^2(\mathtt{H}\circ \mathtt{G})\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right).\hfill \end{array}$$

This completes the proof. □

Theorem 5.

Let $0<{\mathsf{w}}_{\mathsf{1}}<{\mathsf{w}}_{\mathsf{2}},$${\mathtt{G}}_j:\mathbb{A}=[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]\to [0,+\infty )$ be a family of newly proposed definition namely refined convex function of Raina type and $\mathtt{G}\left(u\right)={sup}_j{\mathtt{G}}_j\left(u\right)$. Then $\mathtt{G}$ is refined convex function of Raina type for $m\in (0,1]$, $s\in [0,1]$ and $\lambda \in [0,1],$ and $U=\{\tau \in [{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]:\mathtt{G}\left({\tau }_i\right)<\infty \}$ is an interval.

Proof. 

Let ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in U$, $s\in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $\lambda \in [0,1],$ then

$$\begin{array}{cc}\hfill & \mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & =\underset{j}{sup}{\mathtt{G}}_j({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\hfill \\ & \le \left(1-{\left(s\lambda \right)}^{\alpha }\right)\underset{j}{sup}{\mathtt{G}}_j\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\underset{j}{sup}{\mathtt{G}}_j\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\hfill \\ & =\left(1-{\left(s\lambda \right)}^{\alpha }\right)\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+{\left(1-\left(s\right(1-\lambda \left)\right)\right)}^{\alpha }m\mathtt{G}\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)<\infty .\hfill \end{array}$$

This completes the proof. □

4. Estimations of Hermite–Hadamard Type Inequality

Since the concept of convexity was first proposed more than a century ago, numerous significant inequalities have been presented for the family of convexity. The alleged Hadamard inequality, also known as the Hermite-Hadamard inequality, is the most notable. Hermite and Hadamard introduced this inequality in their ways. It has a variety of applications and an intriguing geometric interpretation. Jensen’s inequality leads to the Hermite-Hadamard inequalities, which are a development of the idea of convexity. It is also quite interesting to note that with the aid of peculiar convex functions, some of the classical inequalities for means can be derived from Hadamard’s inequality. Hermite-Hadamard inequalities for convex functions have attracted a lot of attention lately, leading to an impressive array of improvements and generalizations.

The main focus of this part is to explore and elaborate the Estimations of (H-H) type inequality for refined convex function of Raina type.

Lemma 1.

Let $\mathtt{G}:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$ with ${\mathsf{w}}_{\mathsf{1}}<{\mathsf{w}}_{\mathsf{2}}$. If ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & -\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}+\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx\hfill \\ & =\frac{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{2}{\int }_0^1(1-2\lambda ){\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda .\hfill \end{array}$$

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }\to {\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to given status of $I^{\circ }$.

Integrating by parts implies

$$\begin{array}{cc}\hfill & {\int }_0^1(1-2\lambda ){\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda \hfill \\ & ={\left[\frac{(1-2\lambda )\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\right]}_0^1+\frac{2}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_0^1\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda \hfill \\ & =-\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}+\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx.\hfill \end{array}$$

This led us to the desired proof of Lemma 1. □

Theorem 6.

Let $I^{\circ }$ w.r.t ${\mathcal{F}}_{\rho ,\lambda }^{\sigma }$ is a generalized convex set and $\mathtt{G}:I^{\circ }\subseteq \mathbb{R}\to \mathbb{R}$ be a differentiable mapping on $I^{\circ }$, ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$ with ${\mathsf{w}}_{\mathsf{1}}<{\mathsf{w}}_{\mathsf{2}}$ and suppose that ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$. If $|{\mathtt{G}}^{\prime }|$ is refined convex function of Raina type on $\mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ \hfill & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right)|+m{\mathsf{\Lambda }}_2\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)|\right),\hfill \end{array}$$

holds true for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$, where

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_1\left(s\right)={\int }_0^1|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda =\frac{2{\alpha }^2+6\alpha +4-2.2^{-\alpha +1}{s}^{\alpha }-4\alpha s^{\alpha }}{4(\alpha +1)(\alpha +2)}\hfill \\ \hfill & {\mathsf{\Lambda }}_2\left(s\right)={\int }_0^1|1-2\lambda |(1-{\left(s(1-\lambda )\right)}^{\alpha })d\lambda .\hfill \end{array}$$

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$→${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to the given status of $I^{\circ }$.

Applying Lemma 1, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \left|\frac{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{2}{\int }_0^1(1-2\lambda ){\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda \right|\hfill \end{array}$$

Using refined convex function of the Raina type, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\int }_0^1|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right)|+{(1-\left(s(1-\lambda )\right))}^{\alpha }m\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|)d\lambda \hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}\hfill \\ & \times \left(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right)|{\int }_0^1|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda +m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)|{\int }_0^1|1-2\lambda |{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda \right)\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right)|+m{\mathsf{\Lambda }}_2\left(s\right)\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|\right).\hfill \end{array}$$

This completes the proof. □

Remark 3.

Choosing $\rho =\alpha ,\lambda =1$ with $\sigma =(1,1,\dots )$, and according to the conditions of the above Theorem 6, we attain the inequality in the mode of classical Mittag–Leffler function

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right)|+m{\mathsf{\Lambda }}_2\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)|\right).\hfill \end{array}$$

Theorem 7.

Suppose $I^{\circ }$ be defined in Theorem 6, q > 1, $\frac{1}{p}+\frac{1}{q}=1$ and ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$. If $|{\mathtt{G}}^{\prime }{|}^q$ is refined convex function of Raina type on $\mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\mathsf{\Lambda }}_3\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_4\left(s\right){|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)|}^q\right),\hfill \end{array}$$

holds true for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$, where

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_3\left(s\right)={\int }_0^1(1-{\left(s\lambda \right)}^{\alpha })d\lambda =\frac{\alpha +1-s^{\alpha }}{\alpha +1}\hfill \\ \hfill & {\mathsf{\Lambda }}_4\left(s\right)={\int }_0^1{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda =\frac{-{(1-s)}^{\alpha +1}-1}{s(\alpha +1)}.\hfill \end{array}$$

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$→${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to the given status of $I^{\circ }$.

First, we employing Lemma 1, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \left|\frac{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{2}{\int }_0^1(1-2\lambda ){\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda \right|\hfill \end{array}$$

Using Hölder’s integral inequality, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1{|1-2\lambda |}^p\right)}^{\frac{1}{p}}{\left({\int }_0^1{\left|{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\right|}^{q}d\lambda \right)}^{\frac{1}{q}}\hfill \end{array}$$

Applying refined convex function of the Raina type, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\hfill \\ & \times {\left(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1(1-{\left(s\lambda \right)}^{\alpha })d\lambda +{\int }_0^{1}m{\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda \right)}^{\frac{1}{q}}\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\mathsf{\Lambda }}_3\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_4\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right).\hfill \end{array}$$

This completes the proof. □

Remark 4.

Choosing $\rho =\alpha ,\lambda =1$ with $\sigma =(1,1,\dots )$, and according to the conditions of the above Theorem 7, we attain the inequality in the mode of classical Mittag–Leffler function

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{p+1}\right)}^{\frac{1}{p}}\left({\mathsf{\Lambda }}_3\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_4\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right).\hfill \end{array}$$

Theorem 8.

Suppose $I^{\circ }$ be defined in Theorem 6, $q\ge 1$, and ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$. If $|{\mathtt{G}}^{\prime }{|}^q$ is refined convex function of Raina type on $\mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{1-\frac{2}{q}}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_2\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right),\hfill \end{array}$$

holds true for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$, where ${\mathsf{\Lambda }}_1\left(s\right)$ and ${\mathsf{\Lambda }}_2\left(s\right)$ are defined in Theorem 6.

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$→${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to given status of $I^{\circ }$.

Let $q>1$ and employ Lemma 1, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \left|\frac{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{2}{\int }_0^1(1-2\lambda ){\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))d\lambda \right|\hfill \end{array}$$

Using power mean inequality, we have

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1|1-2\lambda |d\lambda \right)}^{1-\frac{1}{q}}{\left({\int }_0^1|1-2\lambda ||{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})){|}^{q}d\lambda \right)}^{\frac{1}{q}}\hfill \end{array}$$

Finally, we use refined convex function of the Raina type,

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\hfill \\ & \times \left({\int }_0^1|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+{(1-\left(s(1-\lambda )\right))}^{\alpha }{\left|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{1}}\right)\right|}^q\right]d\lambda {)}^{\frac{1}{q}}\hfill \\ \hfill & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{1-\frac{1}{q}}\hfill \\ & \times (|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda +{\int }_0^1|1-2\lambda |m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q(1-{\left(s(1-\lambda )\right)}^{\alpha })d\lambda {)}^{\frac{1}{q}}\hfill \\ & =\frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{1-\frac{2}{q}}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_2\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right).\hfill \end{array}$$

If $q=1$, then we employ the same methodology according to the above Theorem 6. We obtain the required proof of Theorem 8. □

Remark 5.

Choosing $\rho =\alpha ,\lambda =1$ with $\sigma =(1,1,\dots )$, and according to the conditions of the above Theorem 8, we attain the inequality in the mode of classical Mittag–Leffler function

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{1-\frac{2}{q}}\left({\mathsf{\Lambda }}_1\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_2\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right).\hfill \end{array}$$

Theorem 9.

Suppose $I^{\circ }$ be defined in Theorem 6, q > 1, $\frac{1}{p}+\frac{1}{q}=1$ and ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$. If $|{\mathtt{G}}^{\prime }{|}^q$ is refined convex function of Raina type on $\mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}\hfill \\ & \times \left\{{\left({\mathsf{\Lambda }}_5\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_6\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}+{\left({\mathsf{\Lambda }}_7\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_8\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}\right\},\hfill \end{array}$$

holds for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$. Where

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_5\left(s\right)={\int }_0^1(1-\lambda )(1-{\left(s\lambda \right)}^{\alpha })d\lambda =\frac{1}{2}-\frac{s^{\alpha }}{(\alpha +1)(\alpha +2)}\hfill \\ & {\mathsf{\Lambda }}_6\left(s\right)={\int }_0^1(1-\lambda ){(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda =\frac{{(1-s)}^{\alpha +2}-1-({(1-s)}^{\alpha +1}-(\alpha +1))\frac{\alpha +2}{\alpha +1}}{s^2(\alpha +2)}\hfill \\ \hfill & {\mathsf{\Lambda }}_7\left(s\right)={\int }_0^1\lambda (1-{\left(s\lambda \right)}^{\alpha })d\lambda =\frac{\alpha +2-2s^{\alpha }}{2(\alpha +2)}\hfill \\ & {\mathsf{\Lambda }}_6\left(s\right)={\int }_0^1\lambda {(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda =-\frac{{(1-s)}^{\alpha +2}-1-(1-s)[1-s^{\alpha +1}-(\alpha +1)]\frac{\alpha +2}{\alpha +1}}{s^2(\alpha +2)}.\hfill \end{array}$$

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$→${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to the given status of $I^{\circ }$.

First, we Applying Lemma 1,

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ \hfill & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\int }_0^1|1-2\lambda ||{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))|d\lambda .\hfill \end{array}$$

Using Hölder-İscan integral inequality

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1(1-\lambda ){|1-2\lambda |}^{p}d\lambda \right)}^{\frac{1}{p}}{\left({\int }_0^1(1-\lambda ){\left|{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\right|}^{q}d\lambda \right)}^{\frac{1}{q}}\hfill \\ & +\frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1\lambda {|1-2\lambda |}^{p}d\lambda \right)}^{\frac{1}{p}}{\left({\int }_0^1\lambda {\left|{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))\right|}^{q}d\lambda \right)}^{\frac{1}{q}}.\hfill \end{array}$$

Finally, employing refined convex function of Raina type

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}\hfill \\ & \times {\left(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1(1-\lambda )(1-{\left(s\lambda \right)}^{\alpha })d\lambda +{\int }_0^1(1-\lambda )m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda \right)}^{\frac{1}{q}}\hfill \\ & +\frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}\hfill \\ & \times {\left(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1\lambda (1-{\left(s\lambda \right)}^{\alpha })d\lambda +{\int }_0^{1}m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\lambda {(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda \right)}^{\frac{1}{q}}\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}\hfill \\ & \times \left\{{\left({\mathsf{\Lambda }}_5\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_6\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}+{\left({\mathsf{\Lambda }}_7\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_8\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}\right\}.\hfill \end{array}$$

This completes the proof. □

Remark 6.

Choosing $\rho =\alpha ,\lambda =1$ with $\sigma =(1,1,\dots )$, and according to the conditions of the above Theorem 9, we attain the inequality in the mode of classical Mittag–Leffler function

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2(p+1)}\right)}^{\frac{1}{p}}\hfill \\ & \times \left\{{\left({\mathsf{\Lambda }}_5\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_6\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}+{\left({\mathsf{\Lambda }}_7\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_8\left(s\right)|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q\right)}^{\frac{1}{q}}\right\}.\hfill \end{array}$$

Theorem 10.

Suppose $I^{\circ }$ be defined in Theorem 6, $q\ge 1$ and ${\mathtt{G}}^{\prime }\in \mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$. If $|{\mathtt{G}}^{\prime }{|}^q$ is refined convex function of Raina type on $\mathcal{L}[{\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}]$, then

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{2-\frac{2}{q}}\hfill \\ & \times \{{\left({\mathsf{\Lambda }}_9\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{10}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & +{\left({\mathsf{\Lambda }}_{11}\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{12}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\},\hfill \end{array}$$

holds true for $\lambda \in [0,1]$, $(\alpha ,m)\in (0,1]\times (0,1]$ and $s\in [0,1]$. Where

$$\begin{array}{cc}\hfill & {\mathsf{\Lambda }}_9\left(s\right)={\int }_0^1\lambda |1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda ,{\mathsf{\Lambda }}_{10}\left(s\right)={\int }_0^1(1-\lambda )|1-2\lambda |{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda ,\hfill \\ & {\mathsf{\Lambda }}_{11}\left(s\right)={\int }_0^1\lambda |1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda ,{\mathsf{\Lambda }}_{12}\left(s\right)={\int }_0^1\lambda |1-2\lambda |{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda \hfill \end{array}$$

Proof. 

Suppose that ${\mathsf{w}}_{\mathsf{1}},{\mathsf{w}}_{\mathsf{2}}\in I^{\circ }$→${\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})\in I^{\circ }$, because due to given status of $I^{\circ }$.

Let $q>1$. First we applying Lemma 1,

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\int }_0^1|1-2\lambda ||{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))|d\lambda .\hfill \end{array}$$

Using improved power-mean integral inequality

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1(1-\lambda )|1-2\lambda |d\lambda \right)}^{1-\frac{1}{q}}\hfill \\ & \times {\left({\int }_0^1(1-\lambda )|1-2\lambda ||{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})){|}^{q}d\lambda \right)}^{\frac{1}{q}}\hfill \\ & +\frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left({\int }_0^1\left(\lambda \right|1-2\lambda |d\lambda \right)}^{1-\frac{1}{q}}{\left({\int }_0^1\lambda |1-2\lambda ||{\mathtt{G}}^{\prime }({\mathsf{w}}_{\mathsf{2}}+\lambda {\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})){|}^{q}d\lambda \right)}^{\frac{1}{q}}.\hfill \end{array}$$

Applying the refined convex function of Raina type

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{2-\frac{2}{q}}\hfill \\ & \times \left\{\right(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1(1-\lambda )|1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda \hfill \\ & +{\int }_0^1(1-\lambda )|1-2\lambda |m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q{(1-\left(s(1-\lambda )\right))}^{\alpha }d\lambda {)}^{\frac{1}{q}}\hfill \\ & +{\left(|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q{\int }_0^1\lambda |1-2\lambda |(1-{\left(s\lambda \right)}^{\alpha })d\lambda +{\int }_0^1\lambda |1-2\lambda |m|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right){|}^q(1-{\left(s(1-\lambda )\right)}^{\alpha }d\lambda \right)}^{\frac{1}{q}}\}\hfill \\ & \le \frac{|{\mathcal{F}}_{\rho ,\lambda }^{\sigma }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{2-\frac{2}{q}}\hfill \\ & \times \left\{{\left({\mathsf{\Lambda }}_9\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{10}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}+{\left({\mathsf{\Lambda }}_{11}\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{12}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\right\}.\hfill \end{array}$$

If $q=1$, then we employ the same methodology according to the above Theorem 6. We obtain the required proof of Theorem 10. □

Remark 7.

Choosing $\rho =\alpha ,\lambda =1$ with $\sigma =(1,1,\dots )$, and according to the conditions of the above Theorem 10, we attain the inequality in the mode of classical Mittag–Leffler function

$$\begin{array}{cc}\hfill & |\frac{\mathtt{G}\left({\mathsf{w}}_{\mathsf{2}}\right)+\mathtt{G}({\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}}))}{2}-\frac{1}{E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}{\int }_{{\mathsf{w}}_{\mathsf{2}}}^{{\mathsf{w}}_{\mathsf{2}}+E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})}\mathtt{G}\left(x\right)dx|\hfill \\ & \le \frac{|E_{\alpha }({\mathsf{w}}_{\mathsf{1}}-{\mathsf{w}}_{\mathsf{2}})|}{2}{\left(\frac{1}{2}\right)}^{2-\frac{2}{q}}\hfill \\ & \times \{{\left({\mathsf{\Lambda }}_9\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{10}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\hfill \\ & +{\left({\mathsf{\Lambda }}_{11}\left(s\right)|{\mathtt{G}}^{\prime }\left({\mathsf{w}}_{\mathsf{2}}\right){|}^q+m{\mathsf{\Lambda }}_{12}\left(s\right){\left|{\mathtt{G}}^{\prime }\left(\frac{{\mathsf{w}}_{\mathsf{1}}}{m}\right)\right|}^q\right)}^{\frac{1}{q}}\}.\hfill \end{array}$$

Note: We now have comments regarding the comparison of the above refinements. Employing Lemma 1, we examined two theorems, (Theorems 7 and 9), in which we applied the Hölder and its improved version namely Hölder-İscan inequality. Theorem 9 as compared to Theorem 7 provides a good result. Similarly, employing Lemma 1, we examined two theorems (Theorems 8 and 10), in which we employed power mean and its improved version namely improved power means inequality. Theorem 10 as compared to Theorem 8 provides a good result.

5. Conclusions

Currently, the term convex analysis is a very captivating and magnificent field of research interest due to its many potential importance. The term convexity along with the perception of inequalities plays a vital and strong performance in the present-day mathematical investigation. Many mathematicians have explored and enjoyed some new variants of convexity has been stretched out in different modes like quantum calculus, preinvexity, coordinates, fractal sets, fractional calculus, and interval-valued calculus, etc. In this work, we introduced and investigated.

(1)

A novel idea of generalized convex function namely refined convex function of the Raina type.

(2)

Some nice algebraic properties are established via newly examined definition.

(3)

Further, a new lemma is presented.

(4)

Considering this new lemma, several refinements and remarkable extensions of the (H-H) type inequalities are established.

(5)

For the reader’s interest, we add some remarks regarding the Mittag-Leffer function.

(6)

Comparison between the results is investigated.

In the future, we believe that the concept of this work can be led in various modes like time scale calculus, coordinates, fuzzy fractional, interval analysis, fractional calculus, quantum calculus, etc. We imagine that the method and literature of this work will excite the researcher to examine a more remarkable sequel in this field.