HYBRID FRACTIONAL INTEGRAL INEQUALITIES IN 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 two H.H (Hermite–Hadamard)-type inequalities applicable to multiplicative (geometric) convex function via multiplicative [Formula: see text] operators. Subsequently, by leveraging certain characteristics of multiplicative convex functions, we present novel inequalities related to multiplicative convex function via multiplicative [Formula: see text] operators also demonstrating two novel identities applicable to multiplicatively differentiable functions. By leveraging these identities, we then establish inequalities of trapezoid and midpoint types specifically designed for multiplicatively convex functions. Additionally, we explore applications of these findings to special functions and special means.