Approximately two-dimensional harmonic ( p 1 , h 1 ) $(p_{1},h_{1})$ - ( p 2 , h 2 ) $(p_{2},h_{2})$ -convex functions and related integral inequalities

Abstract

Abstract The aim of this study is to introduce the notion of two-dimensional approximately harmonic ( p 1 , h 1 ) $(p_{1},h_{1})$ - ( p 2 , h 2 ) $(p_{2},h_{2})$ -convex functions. We show that the new class covers many new and known extensions of harmonic convex functions. We formulate several new refinements of Hermite–Hadamard like inequalities involving two-dimensional approximately harmonic ( p 1 , h 1 ) $(p_{1},h_{1})$ - ( p 2 , h 2 ) $(p_{2},h_{2})$ -convex functions. We discuss in detail the special cases that can be deduced from the main results of the paper.

Keywords

• Hermite–Hadamard inequality
• Hölder inequality
• Convexity