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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.Mathematics subject classification (1991): 26A51, 26D05, 26D07, 51M16.

Abstract. Given a function f : I → J and a pair of means M and N, on the intervals I and J

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