Novel extensions of k-harmonically convex functions and their applications in information science.

Abstract

Convex analysis theory has found extensive applications in optimization, information science, and economics, leading to numerous generalizations of convex functions. However, a drawback in the vast literature on convex functions is that only a limited number of these notions significantly impact practical applications. With this context, we explore a novel convexity notion known as k-harmonically convex function (k-HCF) using two approaches and present applications in information science. First, we propose an r-parameterized extension of k-HCF, broadening its applicability. Secondly, we extend this concept to interval-valued functions (IVFs), based on a complete order relation on closed bounded intervals. We then investigate properties and inequalities for both extensions to derive lower bounds for information-theoretic measures such as Tsallis entropy, Shannon entropy, and Tsallis relative entropy, using the new parametric extensions of these functions. Additionally, we prove inequalities of the Jensen, Mercer, and Hermite-Hadamard types for the Cr-order-based extension of k-HCFs. Our findings reproduce known results while introducing significant new insights into the field, showing the broader usefulness of k-HCFs in information science.