ON THE GENERALIZED CONVEXITY AND CONCAVITY

Abstract

A function ƒ : R+ → R+ is (m1, m2)-convex (concave) if ƒ(m1(x,y)) ≤ (≥) m2(ƒ(x), ƒ(y)) for all x,y Є R+ = (0,∞) and m1 and m2 are two mean functions. Anderson et al. [1] studies the dependence of (m1, m2)-convexity (concavity) on m1 and m2 and gave the sufficient conditions of (m1, m2)-convexity and concavity of a function defined by Maclaurin series. In this paper, we make a contribution to the topic and study the (m1, m2)-convexity and concavity of a function where m1 and m2 are identric mean, Alzer mean mean. As well, we prove a conjecture posed by Bruce Ebanks in [2].

Keywords

• logarithmic mean
• identric mean
• power mean
• Alzer mean
• convexity and concavity property