Convexity according to the geometric mean

Abstract

We develop a parallel theory to the classical theory of convex functions, based on a change of variable formula, by replacing the arithmetic mean by the geometric one. It is shown that many interesting functions such as exp, sinh, cosh, sec, csc, arc sin, Γ etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of (0, ∞). As a consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones.