FRACTIONAL VERSION OF BULLEN-TYPE INEQUALITIES VIA MULTIPLICATIVE CALCULUS WITH APPLICATIONS

Aspects of both hybrid and fractional calculus are combined in the proportional Caputo-hybrid ([Formula: see text]) operators, which are helpful in solving differential equations with non-integer orders and modeling a variety of complicated phenomena in science and engineering. In this paper, we establish the [Formula: see text] operators via multiplicative calculus which are termed as multiplicative [Formula: see text] operators. We initially formulate Bullen-type ([Formula: see text]-type) identity applicable to multiplicative (geometric) convex function via multiplicative [Formula: see text] operators. Subsequently, leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex functions via multiplicative [Formula: see text] operators. Additionally, we explore applications of these findings to special functions and special means.