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 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.