Novel fractional integral inequalities for GA-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg" display="inline" id="d1e222"><mml:mrow><mml:mi>C</mml:mi><mml:mi>r</mml:mi></mml:mrow></mml:math>-convex functions and connections

Due to applicability of mathematical inequalities in control systems, actuarial science, information theory and its utilization in other sciences, several researchers are inclined to prove their refinements and generalizations. In the study of mathematical inequalities, convex functions (CFs) and the derived classes of CFs are the most studied objects. On the other hand, fractional calculus has emerged as an indispensable tool in Engineering, Physics, and Mathematics, with widespread applications. The recently introduced Cr-order based class of interval-valued functions (IVFs) has shown applications in information theory. In this paper, we aim to investigate mathematical inequalities for geometrically arithmetically Cr-convex functions (GA-Cr-CFs) via interval-valued Hadamard fractional integral operators (IVHFIOs). More precisely, we study the Hermite–Hadamard–Mercer type inequalities and the weighted Hermite–Hadamard–Mercer type inequalities involving GA-Cr-CFs and IVHFIOs. Under particular assumptions, the proved results produce inequalities from the recent related literature. Furthermore, we establish connections between geometrically arithmetically convex functions (GA-CFs) and self information function of each event with a given probability in a complete information system. At the end, we demonstrate the role of fractional order of IVHFIOs in producing the refinements of the inequalities for the usual integrals. Thus, the current study provides both the extension of theoretical literature as well as application aspects.