Exploring properties and inequalities for geometrically arithmetically-Cr-convex functions with Cr-order relative entropy

We introduce a new class of interval-valued functions which we call Geometrically Arithmetically Cr-convex functions (GA-Cr-convex functions) and investigate its properties. We explore necessary and sufficient conditions for an interval-valued function to be a GA-Cr-convex function through two distinct approaches. Firstly, we propose these conditions connecting GA-Cr-convex functions with two real-valued GA-convex functions, and secondly, we examine them with respect to a Crconvex function. Furthermore, we employ our findings to establish generalizations of several renowned inequalities including Jensen's inequality, Jensen-Mercer inequality, and HermiteHadamard inequality for GA-Cr-convex functions. The derived inequalities are consistent with the corresponding inequalities for real-valued functions. Consequently, we re-capture several significant results from recent literature. We also define a strong version of the newly introduced class, investigate its properties and prove well-known inequalities for the strong version as well. Lastly, we demonstrate applications of the Cr-order inequalities in information science by defining Cr-relative entropy via log -sum inequality for interval-valued functions. For Crorder relative entropy, we prove non-negativity, joint convexity and monotonicity to generalize these properties, previously known for Tsallis relative entropy. Consequently, we obtain a generalization of data (information) processing inequality to complete the impact of our work in three directions including generalization, consistency with literature, and applications in information theory.