Structural Properties of Multiplicative Fractional Integral Inequalities

Abstract

This paper studies four qualitative properties of multiplicative fractional integral inequalities: boundedness, sharpness, additivity, and stability. An auxiliary kernel identity is established to express the error between the multiplicative fractional integral mean and the arithmetic-type mean, which is then used with Hölder's and power-mean inequalities to derive sharp two-sided error bounds for multiplicatively convex, Lipschitz-continuous functions, with the extremal function identified explicitly. A semigroup-type additivity result decomposes the error over \([a,b]\) into errors over \([a,c]\) and \([c,b]\) up to a computable correction term, while stability estimates quantify the error's sensitivity to perturbations of \(f\). Corollaries recover known Ostrowski, Simpson, and trapezoid-type results as special cases, and two numerical examples with tables and visualizations confirm all theoretical bounds.