Midpoint Inequalities via Strong Convexity Using Positive Weighted Symmetry Kernels

Abstract

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.

Positive Weighted Symmetry Function Kernels and Some Related Inequalities for a Generalized Class of Convex Functions

“…Fractional calculus is one of the interesting generalizations of classical calculus, as most of the differential equations are fractional in modeling real world problems [1][2][3]. So the fractional integral inequalities are the interesting area of research for many researchers of analysis and inequality [4,5]. The operates involving fractional integrals become a hot area of research for the researcher working in inequality theory (see for example [5] and references therein).…”

“…So the fractional integral inequalities are the interesting area of research for many researchers of analysis and inequality [4,5]. The operates involving fractional integrals become a hot area of research for the researcher working in inequality theory (see for example [5] and references therein). Many fractional integral operators are introduced in recent years, and every operator has its own certain properties, for example, Riemann-Liouville integral operator ( [6,7]), Caputo operator [8], Hadamard operator [9], Caputo Fabrizio operator [10], and Saigo operator [11].…”

“…Definition 1 (see [5,18]). Let ðχ 1 , χ 2 Þ ⊂ R ˇand ℘ðχÞ be a monotone, positive and increasing function defined on ðχ 1 , χ 2 having continuous derivative ℘′ðχÞ on ðχ 1 , χ 2 Þ.…”

“…where I : ½χ 1 , χ 2 ⟶ R ˇis continuous and convex function and ℘ðχÞ is self increasing and monotone on ðχ 1 , χ 2 with ℘′ is continuous function on ðχ 1 , χ 2 Þ and ℏ : ½χ 1 , χ 2 ⟶ ð0,∞Þ is an symmetric integrable function w.r.t ðχ 1 + χ 2 Þ/2, where χ 1 < χ 2 [5]. This paper is organized as follows: initially, we will give some preliminaries material related to our research (Fractional Hermite-Hadamard-Fejer mid-point inequalities for a MT-Convex Function via an increasing function involving a positive weighted symmetric function).…”