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

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.