On multiplicative <i>beta</i> -convexity and associated Katugampola fractional inequalities: Hermite–Hadamard and parametrized Newton–Cotes results

In this work, we introduce a new class of functions, referred to as multiplicative beta-convex functions, which extends several existing convexity concepts in the context of multiplicative calculus. Based on this notion, we establish a Hermite–Hadamard inequality involving multiplicative Katugampola fractional integrals. We also derive a new integral identity associated with these operators, which allows us to construct a unified family of parameterized three-point Newton–Cotes-type inequalities. These inequalities apply to functions whose first-order [Formula: see text]-derivatives, in [Formula: see text]-absolute value, are multiplicative beta-convex. By suitably selecting the parameters, our general framework yields various well-known inequalities as special cases, including midpoint, trapezoidal, Simpson-, and Bullen-type inequalities. These arise naturally under specific subclasses of beta-convexity, such as multiplicative s-, P-, and [Formula: see text]-convexity. To corroborate our theoretical results, we present a numerical example together with graphical illustrations that demonstrate the validity and sharpness of the obtained bounds. Our contributions advance the development of fractional multiplicative analysis and offer new perspectives for applications in numerical approximation and the theory of inequalities.