Convexity according to a pair of quasi-arithmetic means and inequalities

Abstract

We consider a class of generalized convex functions, which are defined according to a pair of quasi-arithmetic means and called (Mϕ,Mψ)-convex functions, and establish various Fejér type inequalities for such a function class. These inequalities not merely provide a natural and intrinsic characterization of the (Mϕ,Mψ)-convex functions, but actually offer a generalization and refinement of the most part of the concrete Hermite-Hadamard and Fejér type inequalities obtained in earlier studies for different kinds of convexity and fractional integrals. Applications to inequalities involving the gamma function and special means are also included.