A Multiplicative Mean Value and Its Applications

We develop a parallel theory to that concerning the concept of integral mean value of a function, by replacing the additive framework with a multiplicative one. Particularly, we prove results which are multiplicative analogues of the Jensen and Hermite-Hadamard inequalities. 1. Introduction Many results in Real Analysis exploits the property of convexity of the subintervals I of R; (A) x; y 2 I; 2 [0; 1] implies (1 )x+ y 2 I which is motivated by the vector lattice structure on R: By evident reasons, we shall refer to it as the arithmetic convexity. As well known, R is an ordered …eld and the subintervals J of (0;1) play a multiplicative version of convexity, (G) x; y 2 J; 2 [0; 1] implies x y 2 J: which we shall refer to as the geometric convexity. Moreover, the pair exp log makes possible to pass in a canonical way from (A) to (G) and vice-versa. As we noticed in a recent paper [8], this fact opens the possibility to develop several parallels to the classical theory involving (A), by replacing (A) by (G) (or, mixing (A) with (G)). The aim of the present paper is to elaborate on the multiple analogue of the notion of mean value. In order to make our de…nition well understood we shall recall here some basic facts on the multiplicatively convex functions i.e., on those functions f : I ! J (acting on subintervals of (0;1)) such that (GG) x; y 2 I and 2 [0; 1] implies f(x y ) f(x) f(y) ; the label (GG) is aimed to outline the type of convexity we consider on the domain and the codomain of f: Under the presence of continuity, multiplicative convexity means f( p xy) p f(x) f(y) for all x; y 2 I which motivates the alternative terminology of convexity according to the geometric mean for (GG). Another equivalent de…nition of the multiplicative convexity (of a functionf) is log f(x) is a convex function of log x: See [8], Lemma 2.1. Modulo 1991 Mathematics Subject Classi…cation. 26A51, 26D07, 26D15. Key words and phrases. Keywords and phrases: Convex function, Mean value, AM GM Inequality. Published in vol. Theory of Inequalities and Applications, vol. 1 (Y. J. Cho, S. S. Dragomir and J. Kim, editors), pp, 243-255, Nova Science Publishers, Huntington, New York, 2001. ISBN 1-59033-188-5. 1 2 CONSTANTIN P. NICULESCU this remark, the class of all multiplicatively convex functions was …rst considered by P. Montel [7], in a beautiful paper discussing the analogues of the notion of a convex function in n variables. As noticed in [8], the class of multiplicatively convex functions contains a broad range of functions from the elementary ones, such as sinh; cosh; exp; on (0;1) tan; sec; csc; 1=x cotx; on (0; =2) arcsin; arccos; on (0; 1] log(1 x); 1 + x 1 x ; on (0; 1) to the special ones, such as j[1;1); Psi;L (the Lobacevski function), Si (the integral sine) etc. The notion of a strictly multiplicatively convex function can be introduced in a natural way and we shall omit the details here. Notice that the multiplicatively a¢ ne functions are those of the form Cx ; with C > 0 and 2 R: Some readers could be frustrated by the status of 0 (of being placed outside the theory of multiplicative convexity). This can be …xed each time we work with functions f such that f(0) = 0 and f(x) > 0 for x > 0. There is a functorial device to translate the results for the (A)-type of convexity to the (G) type and vice-versa, based on the following remark: Lemma 1.1. Suppose that I is a subinterval of (0;1) and f : I ! (0;1) is a multiplicatively convex function. Then F = log f exp : log (I)! R is a convex function. Conversely, if J is an interval and F : J ! R is a convex function, then f = exp F log : exp (J)! (0;1) is a multiplicatively convex function. In the standard approach, the mean value of an integrable function f : [a; b]! R is de…ned by M(f) = 1 b a Z b a f(t) dt and the discussion above motivates for it the alternative notation MAA(f); as it represents the average value of f according to the arithmetic mean. Taking into account Lemma 1 above, the multiplicative mean value of a function f : [a; b]! (0;1) (where 0 < a < b) will be de…ned by the formula MGG(f) = exp 1 log b log a Z log b