New Fractional Hermite–Hadamard–Mercer Inequalities for Harmonically Convex Function

In 2003, Mercer presented an interesting variation of Jensen’s inequality called Jensen–Mercer inequality for convex function. In the present paper, by employing harmonically convex function, we introduce analogous versions of Hermite–Hadamard inequalities of the Jensen–Mercer type via fractional integrals. As a result, we introduce several related fractional inequalities connected with the right and left differences of obtained new inequalities for differentiable harmonically convex mappings. As an application viewpoint, new estimates regarding hypergeometric functions and special means of real numbers are exemplified to determine the pertinence and validity of the suggested scheme. Our results presented here provide extensions of others given in the literature. The results proved in this paper may stimulate further research in this fascinating area.


Introduction
The definition of convexity has been improved, generalized, and expanded in several directions in recent years. In the literature, Jensen's inequality (J-I) and the Hermite-Hadamard's (H-H) inequality are highly familiar results. Several new classes of convex functions along with their respective new variants of (J-I) and (H-H) inequalities are established. One of the well-known and most significant inequalities in mathematical analysis is (J-I) and its related variants. These inequalities are useful in Physics since they provide upper and lower limits for natural phenomena defined by integrals, such as mechanical work. The definition of a classical convex function is as follows: holds provided that all θ, Θ ∈ I and ζ ∈ ½0, 1.
Jensen's inequality is the key to success in extracting applications in information theory. It is effective in finding estimates for several quantitative measures in information theory about continuous random variables, see [1][2][3]. The (J-I) can be stated as a generalization of convex functions as follows: Theorem 2 (see [4]). If ϕ ∈ KðIÞ, then for all a j ∈ I and ω j ∈ ½0, 1, ðj = 1, 2, ⋯, mÞ with ∑ m j=1 ω j = 1.
The (H-H) inequality is another well-known inequality in the theory of convex analysis. Several notable results surrounding (H-H) inequality and its significance are compiled by Dragomir et al. in [5].

Theorem 3.
If ϕ ∈ KðIÞ on the interval I = ½θ, Θ with θ < Θ, then In 2003, Mercer presents a variant of (J-I) which has a great impact on the theory of inequalities known as Jensen-Mercer (J-M) inequality.
In 2013 [7], Kian [8] worked on some improvements and generalization of (J-M) type inequalities. Then in 2020 [9], Adil et al. gave applications of (J-M) inequality in information theory. He computed new estimates for Csiszár and related divergences. Also, he gave new bounds for Zipf-Mandelbrot Entropy via (J-M) inequality.
Harmonic convex sets are introduced by investigating harmonic means. In 2003, the first harmonic convex set was introduced by Shi and Zhang [10]. The harmonic mean has been important in different fields of pure and applied sciences. Anderson et al. [11] and Íşcan [12] introduced a significant class of convex functions known as harmonic convex.
The harmonic mean is useful in electrical circuit theory and different fields of research. It is well known that the total resistance of a set of parallel resistors can be calculated by adding the reciprocals of each individual resistance value and then taking the reciprocal of the total resistance. For example, if t 1 and t 2 are the resistance of two parallel resistors, the total resistance is which is the half of the harmonic mean [13]. The harmonic mean is also important in the development of parallel algorithms for solving nonlinear problems [14]. The harmonic mean of the effective masses, as well as the three crystallographic directions, is often used to describe a semiconductor's "conductivity effective mass" [15]; see also [16]. Dragomir is the first to introduce (J-I) for ϕ ∈ H K ðIÞ as: Theorem 6 (see [17]). If ϕ ∈ H K ðIÞ on the interval I ⊆ ð0,∞Þ, then for all a j ∈ I and ω j ∈ ½0, 1, In [12], Íşcan proved the (H-H) inequality for ϕ ∈ H K ðIÞ as: Theorem 7 (see [12]). Let I ⊆ ð0,∞Þ be an interval. If ϕ ∈ H K ðIÞ and ϕ∈L½θ, Θ and for all θ, Θ∈I with θ < Θ then Very recently, Baloch et al. [18] present a variant of (J-I) which has a great impact on the theory of inequalities known as (J-M) inequality for ϕ ∈ H K ðIÞ:
Let us recall some important functions and inequality.
(i) Beta function (ii) Hypergeometric function: [20] Lemma 9 (see [21,22]). For 0 < α ≤ 1 and 0 ≤ x < y, we have One of the concepts that have played a significant role in the growth of inequality theory in recent years is fractional analysis. Fractional integrals are the most commonly used concept in calculus analysis to obtain new generalizations, extensions, and versions of classical integral inequalities. Since fractional calculus was presented toward the end of the nineteenth century, the subject has become a quickly developing area and has discovered numerous applications in different research fields. Fractional calculus is now concerned with fractional-order integral and derivative operators in real and complex analysis and their applications. Fractional calculus is used in several fields of engineering and science worldwide, including fluid dynamics, electrochemistry, electromagnetics, viscoelasticity, biological population models, optics, and signal processing. It has been used to model physical and engineering processes that are best represented by fractional differential equations. Now, we give the definition of Riemann-Liouville (RL) integrals which we will use in this paper.
In recent times, the topic of investigating fractional (H-H) inequalities by employing the Mercer concept along with its applications is worth study, as evident from several publications in this direction (see [23][24][25][26]). This study is done by utilizing convex functions. But in this paper, we first time introduce and analyze this concept for harmonic convex functions. In this paper, by using (J-M) inequality, we derive Hermite-Hadamard-Mercer's (H-H-M) inequalities for ϕ ∈ H K ðIÞ via (RL) fractional integral, and we established several new fractional inequalities pertaining (H-H-M) type inequalities for differentiable harmonically convex mappings. Some applications to special means of positive real numbers will also be provided in Section 4. We hope that the new idea and techniques formulated in the present paper are more invigorating than the accessible ones.

(H-H-M) Inequalities for ϕ ∈ H K ðIÞ via (RL) Fractional Integrals
By using (J-M) inequality, we give the following (H-H-M) inequalities for ϕ ∈ H K ðIÞ.

Journal of Function Spaces
It is obvious that Using (J-M) inequality for ϕ ∈ H K ðIÞ, we conclude that So, the inequality (15) is proved.☐

Related Variants of (H-H-M) Type Inequalities for ϕ ∈ H K ðIÞ via (RL) Fractional Integrals
Throughout the paper, we assumed the following assumption.

Corollary 14.
If we choose α = 1 in Lemma 13, then we have the following equality: Remark 15. If we take x = θ and y = Θ in Corollary 14, then the equality (34) reduces to the equality which is proved by Íşcan in [12].
Using Lemma 13, we present the following fractional integral inequality for jϕ ′ j q ∈ H K ðIÞ as follows.
When 0 < α ≤ 1, using Lemma 9 and Lemma 13, we can obtain another results for jϕ′j q ∈ H K ðIÞ via fractional integral as follows.

Proof. Let
Calculating Z 4 and Z 5 , we have Using (51) and (52) in (50), we get the inequality of (49). This completes the proof.☐ Remark 22. If we take x = θ and y = Θ in Theorem 21, then it becomes Theorem 7 proved by Íşcan and Whu in [27].

Conclusion
In this paper, we present the (H-H-M) inequalities involving (RL) fractional integrals for the class of harmonic convex function (instead of convex function) and established some integral inequalities connected with the right and left sides of fractional (H-H-M) type inequalities for differentiable mappings whose derivatives in absolute value are harmonically convex. Some applications to special means have also been presented. Our obtained results are an extension of previously known results. An interesting topic is whether we can use the techniques in this paper to establish the (H-H-M) inequalities for other kinds of convex functions via (RL) fractional integrals. Our ideas and approach may stimulate further research for the researchers working in this field.

Data Availability
Data sharing does not apply to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest
The authors declare that they have no conflicts of interest.