Integer and Fractional Order Inequalities for (P, m)-Convex Functions in the Context of Multiplicative Calculus

Abstract

In this work, we introduce a novel family of functions known as multiplicatively (P, m) convex functions, which generalize the classical concept of convexity in the multiplica tive sense. We rigorously study the properties of these functions, focusing on their struc tural characteristics and behavior. As part of this exploration, we establish integer-order Hermite-Hadamard-type inequalities, which provide new insights into the integral bounds for such functions. Building on this framework, we introduce the concept of multiplicative k-Riemann Liouville (k-RL) fractional integrals, extending the classical integral operators into the fractional domain. We thoroughly investigate their fundamental properties, including inte grability, continuity, and commutativity, demonstrating their robustness and utility in ad vanced mathematical analysis. Leveraging these operators, we derive several novel k-RL fractional Hermite-Hadamard-type inequalities and Newton-type inequalities tailored to multiplicatively (P, m)-convex functions. To validate the correctness and applicability of the proposed results, we present a series of detailed examples alongside their graphical representations, which provide visual con f irmation of the derived inequalities. These examples illustrate the practical implications of the theoretical findings and enhance the understanding of the results. Moreover, we explore real-world applications of these inequalities in the context of special means, including arithmetic, geometric, and harmonic means. The inequalities are further applied to analyze optimization problems and study the properties of certain classes of special functions. This includes applications in information theory, where the derived inequalities can be utilized to estimate bounds for divergence measures and entropy-like functionals. The framework can also be extended to analyze systems governed by stochas tic processes and fractional calculus in applied sciences. Theresults presented in this work not only extend the theoretical foundations of convex ity and fractional calculus but also offer practical tools for researchers in pure and applied mathematics, particularly in areas involving optimization, numerical analysis, and mathe matical modeling.