Generalized convexity and inequalities

Let R+ = (0,∞) 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 m1,m2 ∈M, we say that a function f :R+→R+ is (m1,m2)-convex if f (m1(x, y)) m2(f (x), f (y)) for all x,y ∈ R+. The usual convexity is the special case when both mean values are arithmetic means. We study the dependence of (m1,m2)-convexity on m1 and m2 and give sufficient conditions for (m1,m2)-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. © 2007 Elsevier Inc. All rights reserved.