Weighted Hermite–Hadamard integral inequalities for general convex functions

: In this article, starting with an equation for weighted integrals, we obtained several extensions of the well-known Hermite–Hadamard inequality. We used generalized weighted integral operators, which contain the Riemann–Liouville and the k -Riemann–Liouville fractional integral operators. The functions for which the operators were considered satisfy various conditions such as the h -convexity, modiﬁed h -convexity and s -convexity.


Introduction
Discussing convexity in the field of mathematics is immersed in the realm of interdisciplinary relations.Convexity serves as a foundational concept not only within the domains of geometry and analysis, but also finds extensive applications across various diverse fields.The utility of convexity techniques permeates numerous branches of mathematics, encompassing both pure and applied disciplines.These include optimization, the theory of inequalities, functional analysis, mathematical programming, game theory, number theory, variational calculus and their intricate interplay.This continual fusion of convexity concepts with other mathematical realms consistently yields fertile ground for the inception of novel research endeavors and the development of practical applications.Thus, it becomes evident that the concept of convex functions plays an indispensable and central role in the landscape of contemporary mathematics.

A real function
holds for all y, x ∈ [ 1 , 2 ] and 0 ≤ ξ ≤ 1.If the above inequality is reversed, then function Φ is said to be concave on [ 1 , 2 ].
Convex functions have been studied extensively.Moreover generalizations have been considered, including the n-convex, r-convex, m-convex, s-convex, modified h-convex and (h, m)-convex functions and numerous others (a much more general overview of the different notions of convexity can be found in [1]).
The following definitions favor the reading of our work.
Definition 1.1.[2] If a function Φ : I → R satisfies the conditions of being nonnegative and for all x, y ∈ I and 0 ≤ ξ ≤ 1, the inequality Φ(ξx + (1 − ξ)y) ≤ Φ(x) + Φ(y) holds, then it can be classified as a member of the set P(I).
[3] For some fixed s from the interval (0, 1], a real function Φ given in [0, ∞) is defined as s-convex in the second sense if the condition is satisfied for all x, y in the domain [0, ∞) and for any ξ in the interval (0, 1).
[4] Consider a nonnegative function h : J → R, where h is not equal to zero.We designate a function Φ : I → R as an h-convex function, or state that Φ belongs to the class S X(h, I), under the condition that Φ is nonnegative and for all x, y in the domain I, and ξ in the open interval (0, 1), the following inequality holds: If inequality (1.1) is reversed, then Φ is said to be h-concave.Clearly, if h(ξ) = ξ, then we have the classic convex, if h(ξ) = 1, then we have the P-functions and if h(ξ) = ξ s , where s ∈ (0, 1], then we obtain the s-convex functions of second sense.
If it satisfies the condition that for all x, y ∈ [0, 2 ] and 0 ≤ ξ ≤ 1, the inequality below holds: In our work, we will use the following definition.
One of the most significant inequalities associated with the concept of convexity, which has garnered the attention of inequality experts over recent decades, is the renowned Hermite-Hadamard inequality: This inequality applies to any function f that is convex over the interval [a, b].Hermite first published this inequality in 1883, and, independently, Hadamard presented it in 1893.The Hermite-Hadamard inequality not only offers an estimation for the average value of a convex function, but also serves as an refinement to the Jensen inequality.If the reader wishes to delve deeper into this topic and explore further extensions of the Hermite-Hadamard inequality, we recommend the cited sources [7][8][9][10][11][12][13][14][15][16] and the references contained therein.
, then the left and right k-Riemann-Liouville fractional integrals of order α > 0 are defined by Now, we introduce the integral operators with weights, which will serve as the foundation of our research.
Then, the weighted fractional integrals are defined by (left and right, respectively): , then we obtain the Riemann-Liouville fractional integrals of order α > 0, left and right, respectively.Other fractional integral operators, such as k-Riemann-Liouville fractional integrals, can be easily obtained with the proper selection of w (ξ).
In this study, we introduce several variants of inequality (1.2) within the context of the weighted integral operators as defined in Definition 1.8.

Results
The following Lemma is a basic result of our work.
, then we have the following equation: (2.1) By integrating each of these integrals two times using integration by parts, we achieve hence, we get From this last equality, with changes of variables , respectively and after some algebraic technicality, the desired result is obtained.Our first two fundamental results are the following.
Remark 2.2.If we consider w (ξ) ≡ 1, h(ξ) = ξ, from the previous result, we obtain the following estimate of the left hand side of the Hermite-Hadamard inequality (1.2): This inequality was obtained in several previous papers, for example in [22, inequality (15)] and in [23, Proposition 1].
Theorem 2.6.Given the conditions outlined in Lemma 2.1, in the event that |Φ | q is h-convex over the interval [ 1 , 2 ], for q > 1 the subsequent inequality holds: Proof.The proof is analogous to that of Theorem 2.5.We omit the details.

Conclusions
In this work we have obtained some inequalities using a certain weighted integral, which contained several already published results.Apart from the remarks made, we can point out the strength of our approach due to the fact that we considered general convex functions such as modified h-convex, s-convex or h-convex functions.Consider the following example.Moreover, we can cover some known results other than the above remarks.The following is sufficient as an example.Consider the continuous function w : [0, 1] → [0, ∞) with first and second order derivatives piecewise continuous on [0, 1] so that w(0) = w(1) = 0, then we can formulate the following result that can be proved very similarly to Lemma 2.1.This outcome includes a specific instance from [24, Lemma 1] (also discussed in [9]) by setting w(ξ) = ξ(1 − ξ).
We can derive alternative formulations of our findings by pursuing two directions: first, by introducing supplementary constraints on the function w, and second, by exploring alternative concepts of convexity.