Convexity according to the geometric mean

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, 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.