Generalized Convexity and Inequalities

Details Generalized Convexity and Inequalities Generalized Convexity and Inequalities

2007-01-09

arxiv.org/abs/math/0701262v1

J. Math. Anal. Appl. 335 (2007), 1294-1308

Mathematics

Classical Analysis and ODEs

17 pages

Scientific paper

10.1016/j.jmaa.2007.02.016

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 m1, m2 in M, we say that a function f : R+ to R+ is (m1,m2)-convex if f(m1(x,y)) < or = m2(f(x),f(y)) for all x, y in 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.

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