Midpoint Inequalities via Strong Convexity Using Positive Weighted Symmetry Kernels

Party School of Shandong Provincial Committee of the Communist Party of China (Shandong Administration College), Jinan 250014, China The Research Center of Theoretical System of Socialism with Chinese Characteristics in Shandong Province, Jinan 250014, China Department of Mathematics, Government College University, Lahore 54000, Pakistan Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand


Introduction
One of the most interesting research areas of classical analysis is the study of functions and operators, especially convex functions, due to its applications in both integration and differentiation. In the last few years, a great effort has been put to develop new inequalities in convex analysis to deal with the various new applications, since modern problems are modeled by fractional calculus and new applications. So, the classical convexity [1,2] and its related [3,4] inequalities are not enough to tackle these ones. The new fractional integral inequalities in convexity are always appreciable. Moreover, the generalized and new mode of convexity is the area of interest for most of researcher of convex analysis [5,6].
A function f : I ⟶ R is said to be convex on I, if the inequality holds for all r, s ∈I and ϑ ∈ ½0, 1, Many problems may discuss in convexity of sets and functions. In recent year, convexity of sets and functions has been main object of study [7][8][9]. Some new generalized ideas in this point of view are pseudoconvex function, strongly convex function, quasiconvex, generalized convex function, preinvex functions [10], B-convex function, and invex functions. There are many different fundamental books of convex analysis optimization [11,12].
Fractional calculus [13,14] is not a new concept in mathematics, and similar discussion and controversy are observe in history by famous mathematician like Jensen, Hermite, H older, and Stolz. However, the subject of fractional calculus from an applied point of view got rapid development last years. Like other fields of mathematics, this also influences the integral inequalities and convex analysis ( [15]). As a result, various trends in the result are settled recently. The famous fractional integral operators involve Riemann-Liouville [16], Caputo [17,18], Hadamard [19], and Caputo Fabrizio [20,21]. For more details about fractional integral operators, we refer [22][23][24].
The classical Hermite-Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation, and it has many applications [25][26][27]. Recall Hermite-Hadamard-type inequality (simply H-H type inequality) which is given as: Suppose function s : ½c, d ⊂ R is convex, and then the inequality is called the Hermite-Hadamard Inequality.
In the present research, we generalize the midpoint inequalities for strongly convex functions in weighted fractional integral settings. Our results generalize many existing results and can be considered as extension of existing results.

Definitions and Basic Results
Definition 1. Assume that j ⊆ R is an interval and that "a" is a positive integer. If a function s : j ⊂ R ⟶ R is strongly convex with modulus a, it is called strongly convex with modulus a.
Adamek expanded on the idea of a strongly convex function. They replaced the nonnegative term with a real-valued nonnegative function and defined it as follows: If a function is strongly convex, it is defined as such.
where t is the integrable function.
Endpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals, respectively, in [22,24].
where s is the positive convex function, continuous on the closed interval ½d 1 , Then, left-and right-sided weighted fractional integral of a function s, according to another function σðβ 1 Þ on ½d 1 , d 2 , is defined by [25]: where Midpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals and weighted fractional integrals with positive weighted symmetric function in a kernel, due to using the midpoint d 1 + m/2 of the interval given by, respectively, in [33,34].
By making use of (20) and (21) in (18), we get the desired result.
On the other hand, we can prove the second inequality of Theorem 6 by making use of the strong convexity of "s" to get Multiplying both sides of above equation (16) by l v−1 ψð ðl/2Þd 1 + ð2 − l/2Þd 2 Þ and integrating the resulting inequality with respect to l over [0, 1] get Now, by using ψððl/2Þd (21) in (23), we get This ends our proof.
Remark 7. From Theorem 6, we can obtain some special cases as follows: (1) If a = 0, σðβ1Þ = x, then inequality (14) becomes Journal of Function Spaces s ′ ðlÞdl, and let ψ : ½d 1 , d 2 ⟶ R be an integrable, positive, and weighted symmetric function with respect to d 1 + d 2 /2. If σ is a continuous increasing mapping from the interval ½d 1 , d 2 Þ onto itself with a derivative σ′ðβ 1 Þ which is continuous on ðd 1 , d 2 Þ, then for n > 0, the following equality is valid:

Main Results
In this section, by using Lemma 8, one can extend to some new H-H-F type inequalities for strong convex functions. be an integrable, positive, and weighted symmetric function with respect to d 1 + d 2 /2. If in addition, js ′ j is strong convex on ½d 1 , d 2 , and σ is an increasing and positive function from ½d 1 , d 2 onto itself such that its derivative σ ′ ðβ 1 Þ is continuous on ðd 1 , d 2 Þ; then for v > 0, the following inequalities hold: Proof. By making use of Lemma 8 and properties of the modulus, we obtain Since |s ′ | is strongly convex on ½d 1 , d 2 , we get for l ∈ ½σ −1 ðd 1 Þ, σ −1 ðd 2 Þ:

Journal of Function Spaces
Hence, we obtain By putting value of integration in above inequalities, and simple computations yield which complete our proof.