On New Refinement of Convexity via Interval Calculus with its Fractional Extensions with Applications in Information Theory

Abstract

This paper introduces, for the first time, a novel class of center-radius order P-superquadratic interval-valued functions and systematically examines their fundamental structural properties. Leveraging these properties, we establish new Jensen and Hermite-Hadamard (\(\mathcal{H.H}\)) type inequalities, together with their fractional extensions via Riemann-Liouville (\(\mathcal{R.L}\)) fractional integral operators within the framework of interval calculus. The validity and sharpness of our results are demonstrated through numerical examples and graphical illustrations. Furthermore, we enrich the theoretical developments with applications in information theory, yielding significant generalizations and improvements over several existing results. Future research directions include extending these inequalities to other fractional operators, exploring multidimensional settings, and investigating potential applications in optimization and machine learning under uncertainty.