A STUDY ON NEWTON-TYPE INEQUALITIES BOUNDS FOR TWICE *DIFFERENTIABLE FUNCTIONS UNDER MULTIPLICATIVE KATUGAMPOLA FRACTIONAL INTEGRALS

In this study, we are particularly drawn to investigating Newton-type inequalities for twice *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 ${\mathrm{{\rm Y}}}^{\ast \ast }$ being multiplicatively convex or ${(\mathrm{\text{ln}}{\mathrm{{\rm Y}}}^{\ast \ast })}^v$ being convex for $v>1$, with a specific concentration on the case where $0<v\le 1$. 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.