I.V-<inline-formula><math display="inline"><semantics><mi mathvariant="script">CR</mi></semantics></math></inline-formula>-<i>γ</i>-Convex Functions and Their Application in Fractional Hermite–Hadamard Inequalities

Abstract

In recent years, the theory of convexity has influenced every field of mathematics due to its unique characteristics. Numerous generalizations, extensions, and refinements of convexity have been introduced, and one of them is set-valued convexity. Interval-valued convex mappings are a special type of set-valued maps. These have a close relationship with symmetry analysis. One of the important aspects of the relationship between convex and symmetric analysis is the ability to work on one field and apply its principles to another. In this paper, we introduce a novel class of interval-valued (I.V.) functions called CR-γ-convex functions based on a non-negative mapping γ and center-radius ordering relation. Due to its generic property, a set of new and known forms of convexity can be obtained. First, we derive new generalized discrete and integral forms of Jensen’s inequalities using CR-γ-convex I.V. functions. We employ this definition and Riemann-Liouville fractional operators to develop new fractional versions of Hermite-Hadamard’s, Hermite-Hadamard-Fejer, and Pachpatte’s type integral inequalities. We examine various key properties of this class of functions by considering them as special cases. Finally, we support our findings with interesting examples and graphical representations.

Keywords

• interval-valued functions
• Hermite-Hadamard’s inequality
• Jensen’s inequality
• CR -γ-convex interval-valued functions%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> <named-content content-type="font-family:bold-italic"><inline-formula> <mml:math id="mm11113"> <mml:semantics> <mml:mi mathvariant="script">CR</mml:mi> </mml:semantics> </mml:math> </inline-formula></named-content>-<i>γ</i>-convex interval-valued functions