A STUDY ON NEWTON-TYPE INEQUALITIES BOUNDS FOR TWICE ∗DIFFERENTIABLE FUNCTIONS UNDER MULTIPLICATIVE KATUGAMPOLA FRACTIONAL INTEGRALS
Dingyi Ai, Tingsong Du
Abstract
In this study, we are particularly drawn to investigating Newton-type inequalities for twice [Formula: see text]differentiable functions, which are based on multiplicative Katugampola fractional integrals. Toward this goal, we introduce a multiplicative Katugampola fractional identity, forming the basis upon which we establish a sequence of Newton-type inequalities. The derivation of these inequalities is conditioned on [Formula: see text] being multiplicatively convex or [Formula: see text] being convex for [Formula: see text], with a specific concentration on the case where [Formula: see text]. To help readers fully comprehend the results, we provide illustrative examples and corresponding graphs that validate the derived inequalities. Finally, we showcase the applications of the obtained inequalities in quadrature formulas and special means.