The Hermite–Hadamard–Mercer Type Inequalities via Generalized Proportional Fractional Integral Concerning Another Function

In order to be able to study cosmic phenomena more accurately and broadly, it was necessary to expand the concept of calculus. In this study, we aim to introduce a new fractional Hermite–Hadamard–Mercer’s inequality and its fractional integral type inequalities. To facilitate that, we use the proportional fractional integral operators of integrable functions with respect to another continuous and strictly increasing function. Moreover, we establish some new fractional weighted φ -proportional fractional integral Hermite–Hadamard–Mercer type inequalities. Furthermore, in this article, we are keen to present some special cases related to our current study compared to the previous work of the inequality under study.


Introduction
ere is no doubt that a researcher in the field of calculus knows the significant importance that fractional calculus has acquired recently due to its multiple and important uses in many fields in the natural sciences and technology, especially in physics, fluid dynamics, biology, image processing, control theory, computer networking, and signal processing.
Fractional calculus is the generalized form of classical integrals and derivatives for the order is a noninteger, which comes within the framework of mathematicians' relentless pursuit of developing mathematics to make it more general and useable in most cases that may encounter when studying and analyzing natural phenomena. According to this, we can say fractional calculus has become the focus of a large number of researchers' attention. As a result, a lot of extensions and generalizations have appeared especially on the classical fractional calculus like the definitions of Riemann-Liouville (RL) and Caputo. Actually, the derivative Riemann-Liouville is the most general concept and the most uniform and natural. In general, there are numerous other definitions of fractional operators such as Erdélyi-Kober, Hilfer, Katugampola, Hadamard, and Riesz which are just a few examples to make reference to [1,2]. It should be noted that there are many modern fractional operators proposed by many researchers and perhaps the most prominent of them is the recently proposed ABC operator by Atangana and Baleanu [3,4].
holds for all r, s ∈ [a, z] and η ∈ [0, 1]. We say that g is a concave function if inequality (1) is reversed. In general, the real-valued function g is said to be a convex function on [a, z] if and only if for all y 1 , y 2 , . . . ., y n ∈ [a, z] and for any η i ∈ [a, z], i � 1, 2, . . . ., n with i�n i�1 η i � 1, we have is well-known inequality is called Jensen inequality [5].
Convexity of functions with their features is one of the most useful properties among other categories of functions in the important fields of applied sciences, especially statistics and mathematics, which according to its own useful definition has a geometric interpretation. Furthermore, it is a vital part of inequalities theory and has become the leading point for creating numerous inequalities such as Jensen's inequality, Hadamard's inequality with its type inequalities, and Steffensen's inequality. One of these inequalities that are closely related to the convexity of functions is the Hermite-Hadamard inequality, which has a well-known area in the space of inequalities theory.
is inequality was initially proposed by Hermite in 1881, but it did not come into prominence until it was enriched by Hadamard in 1893 [6] as follows: where g is a convex function on [a, z], which is called the Hermite-Hadamard (H-H) inequality. Significantly, H-H inequality recently has become the focus of attention of several mathematicians and researchers due to its remarkable applications and its spacious uses in several diverse areas. Concerning that, a large number of articles have appeared that contain extensions and generalizations of this inequality (see [7][8][9][10][11][12]).
Lemma 1 (see [13] In general, McD Mercer in [13] proved the following generalization of each of inequality (2) and inequality (4), which is called the well-known Jensen-Mercer inequality.
Inequality (5) is a matter of supreme interest due to much information and its explicit boundary conditions. In mathematics and engineering sciences, Jensen-Mercer's inequality and associated inequalities have wonderful applications and their generalizations and extensions have been an excellent topic of research for mathematicians and authors in the past few years as seen through a variety of investigations on the subject. Moradi and Furuichi [14] (2020) presented some new generalizations and improvements of Jensen-Mercer's type inequalities. Khan et al. [15] (2020) applied Jensen-Mercer's inequality in information theory to compute new ratings for Csiszár and related divergence. For more generalizations and details of Jensen-Mercer's type inequalities, see [16][17][18].
Many researchers, motivated by all the above literature, did a lot of research and were able to derive a new inequality which is a mixture of H-H and Jensen-Mercer's inequalities, which was named the Hermite-Hadamard-Mercer's inequality which is our focus through introducing this article.

Theorem 2.
Let g: [a, z] ⟶ R be a convex function. e following inequalities hold: The same authors, in the same work, presented the following inequality. Theorem 3. Let g: [a, z] ⟶ R be a convex function. e following inequalities hold: Iscan [20] (2020) employed the RL fractional integral to investigate some weighted Hermite-Hadamard-Mercer's type inequalities as follows.
Iscan also, in the same work, gave the following weighted Hermite-Hadamard-Mercer's inequality.
All of what we mentioned above prompts us to study Hermite-Hadamard-Mercer's inequality via the recently generalized operators. Here, in this study, we aim to establish Hermite-Hadamard-Mercer's inequality and its type inequalities for convex functions employing proportional fractional integral operators involving continuous strictly increasing functions. We also aim to present some fractional weighted Hermite-Hadamard-Mercer type inequalities via the current generalized integral operators. Along with this study, we are able to discuss some special cases and some relationships between our current study and previous studies. e organization of this research paper will be as follows: In Section 2, we will mention some notations, definitions, and preparatory acquaintance which are used in this work. Section 3 is devoted to the first part of our major results which contain Hermite-Hadamard-Mercer's inequalities.

Essential Preliminaries
Here, we characterize some of the basic properties and some definitions of several elementary fractional integral operators which include the final generalized fractional operator we used to obtain and discuss our new results.
Definition 2 (see [1]). Suppose that the function g is integrable on [a, z] and a ≥ 0. en, for all β > 0, we have where . e notations I β a + g(y) and I β z − g(y) are called, respectively, the left-and right-sided Riemann-Liouville fractional integrals of a function g for the order β.
Definition 3 (see [1,2]). Suppose that the function g is integrable on the interval 5, and let φ be an increasing function, where φ(y) ∈ C 1 (5, R) such that φ ′ (y) ≠ 0 and y ∈ 5. en, for all β > 0, we have e notations φI β a + g(y) and φI β z − g(y) are, respectively, called the left-and right-sided φ-Riemann-Liouville fractional integrals of a function g for the order β.
Definition 4 (see [29]). For the function g, let δ > 0, and we have for all β ∈ C and Re(β) ≥ 0, where International Journal of Mathematics and Mathematical Sciences are, respectively, called the left-and right-sided proportional fractional derivatives of a function g for the order β.
Definition 5 (see [29]). For the integrable function g, let δ > 0, and we have for all β ∈ C and Re(β) ≥ 0, e notations (I β,δ a + g)(y) and (I β,δ z − g)(y) are, respectively, called the left-and right-sided proportional fractional integrals of a function g for the order β.
Definition 6 (see [30]). For the integrable function g and for the strictly increasing continuous function φ on [a, z], let δ ∈ (0, 1], and we have for all β ∈ C and Re(β) ≥ 0, where are, respectively, called the left-and right-sided proportional fractional derivatives of a function g with respect to φ for the order β.
Definition 7 (see [30]). For the integrable function g and for the continuous and strictly increasing function φ on [a, z], let δ ∈ (0, 1], and we have for all β ∈ C and Re(β) ≥ 0, e notations (φI β,δ a + g)(y) and (φI β,δ z − g)(y) are, respectively, called the left-and right-sided proportional fractional integrals of a function g with respect to φ for the order β.

Fractional Hermite-Hadamard-Mercer Inequalities Involving φ-Proportional Fractional Integrals
is section is the first part of our main contributions. Here, we present basic generalization in Hermite-Hadamard-Mercer's inequalities which involve convex functions for generalized proportional fractional integral operators concerning another strictly increasing continuous function.
Proof. According to the Jensen-Mercer inequality and for r, s ∈ [a, z], we have Now, we change the variables r and s with r � ηx + (1 − η) y and s � (1 − η)x + ηy, and we get which leads to On both sides of (36), taking product by exp[δ − 1/δη(y − x)]η β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we obtain Using identity (31) on both sides of (37), we obtain International Journal of Mathematics and Mathematical Sciences 5 β g a + z − x + y 2 Next, Putting φ(u) � ηx + (1 − η)y and φ(v) � (1 − η)x + ηy, we get is proves the first inequality in (32). To prove the second inequality and by using the convexity of g, we can be certain that en, for φ(u) � ηx + (1 − η)y and φ(v) � (1 − η)x + ηy, we have On both sides of (42), taking product by exp[δ − 1/δη(y − x)]η β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we get On both sides of inequality (44), adding g(a) + g(z), we get the second inequality in (32). Hence, desired inequality (32) is thus proved. We now give the proof of inequalities in (33). We have, according to the convexity of the function g, for all r, s ∈ [a, z], that By applying the change of variables a On both sides of (46), taking product by exp[δ − 1/δη(y − x)]η β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we obtain Next, (48)

International Journal of Mathematics and Mathematical Sciences
Putting is completes the proof of the first inequality in (33). To prove the second inequality and by using the convexity of g, we can be certain that Adding inequalities (50) and (51), we obtain On both sides of (52), taking product by exp[δ − 1/δη(y − x)]η β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we obtain On the left-hand side in (53), applying the same arguments as above, we obtain 8 International Journal of Mathematics and Mathematical Sciences which is the second and third inequalities in (33). Hence, the desired inequalities in (33) are thus proved.

Remark 1
e proportional fractional integral version of eorem 6 was provided by K. Yildirim and S. Yildirim in [32].

International Journal of Mathematics and Mathematical Sciences
Proof. According to the convexity of the function g for all r, s ∈ [a, z], we have Putting r � η/2x + 2 − η/2y and s � 2 − η/2x + η/2y, it follows, for all r, s ∈ [a, z] and η ∈ [0, 1], that On both sides of (61), taking product by exp[δ − 1/δη/2(y − x)](η/2) β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we obtain Next, Putting is proves the first inequality in (59). To prove the second inequality and by using the Jensen-Mercer inequality, we can be certain that Adding inequalities (65) and (66), we get On both sides of (67), taking product by exp[δ − 1/δη/2(y − x)](η/2) β− 1 and then integrating the estimating inequality with respect to η over [0, 1], we obtain By comparing the left-hand side of inequality (64) with the left-hand side of inequality (68), we can deduce which is the second inequality in (59). e proof is thus completed.

Weighted Fractional Hermite-Hadamard-Mercer Inequalities Involving φ-Proportional Fractional Integrals
is section is the second part of our main contributions, within which we give the fractional weighted Hermite-Hadamar-Mercer's inequalities which involve convex functions for generalized proportional fractional integral operators concerning another strictly increasing continuous function.

Conclusion
In view of the significant importance recently achieved by fractional calculus and its very important applications in the interpretation and modeling of natural phenomena, it has become necessary to develop and refine our capabilities to generalize some of the recent results related to this topic. We achieved our goals of introducing a new fractional Hermite-Hadamard-Mercer's inequality and its fractional integral type inequalities by employing the proportional fractional operators of integrable functions with respect to another continuous and strictly increasing function. We enhanced our work by establishing some new fractional weighted φ-proportional fractional integral Hermite-Hadamard-Mercer type inequalities. Also, in this article, we were keen to present some special cases related to our current study compared to the previous work of the inequality under study. In future work, we recommend researchers study the current inequality via recent fractional operators such as the Atangana + Baleanu operator or Caputo + Fabrizio operator.
Data Availability e data analysis in this article is all theory.

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