Generalized convexity and inequalities

Let R+ = (0, infinity) and let M be the family of all mean values of two numbers in R+ (some examples are the arithmetic, geometric, and harmonic means). Given in m(1), m(2) is an element of M, we say that a function f : R+ -> R+ is (m(1), m(2))-convex if f(m(1)(x, y)) <= m(2)(f (x), f (y)) for all x, y is an element of R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m(1), m(2))-convexity on m(1) and m(2) and give sufficient conditions for (m(1), m(2))-convexity of functions defined by Maclaurin series. The criteria involve the Maclaurin coefficients. Our results yield a class of new inequalities for several special functions such as the Gaussian hypergeometric function and a generalized Bessel function. (c) 2007 Elsevier Inc. All rights reserved.