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 Γ etc illustrate the multiplicative version of convexity when restricted to appropriate subintervals of 0 .A s a consequence, we are not only able to improve on a number of classical elementary inequalities but also to discover new ones.