On generalization of Hermite–Hadamard–Mercer inequalities for interval-valued functions with generalized geometric–arithmetic convexity

Abstract

In this paper, we introduce a new convexity notion for inter-valued functions, known as Geometrically–Arithmetically Cr-[Formula: see text]-convex functions (abbreviated as GA-Cr-[Formula: see text]-CFs) and explore its properties. The family of GA-Cr-[Formula: see text]-CFs simultaneously covers the family of GA-CFs, GA-[Formula: see text]-CFs and GA-Cr-CFs. On the other hand, we demonstrate its equivalence with convex functions (CFs), [Formula: see text]-CFs, Cr-CFs and Cr-[Formula: see text]-CFs via exponential function. We also investigate necessary and sufficient conditions for GA-Cr-[Formula: see text]-CFs via two corresponding real-valued GA-[Formula: see text]-CFs. Due to significance and applications of mathematical inequalities, we apply our findings and propose several inequalities such as Jensen–Mercer inequality (JMI), Hermite–Hadamard inequality (HHI) and weighted Hermite–Hadamard-type inequalities (wHHIs) for GA-Cr-[Formula: see text]-CFs. Through our results, we re-capture many known results and inequalities for subclasses of GA-Cr-[Formula: see text]-CFs studied in the recent literature.