NEW MULTI-FUNCTIONAL APPROACH FOR κTH-ORDER DIFFERENTIABILITY GOVERNED BY FRACTIONAL CALCULUS VIA APPROXIMATELY GENERALIZED (ψ, ℏ)-CONVEX FUNCTIONS IN HILBERT SPACE

Abstract

This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized [Formula: see text]-convex and approximately [Formula: see text]-quasiconvex function, with respect to Raina’s function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized [Formula: see text]-convex functions such as higher-order strongly (HOS) generalized [Formula: see text]-convex functions and HOS generalized [Formula: see text]-quasiconvex function. The core of the proposed method is a newly developed Simpson’s type of identity in the settings of Riemann–Liouville fractional integral operator. Based on the HOS generalized [Formula: see text]-convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized [Formula: see text]-quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields.