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arXiv · math/0701262

Generalized Convexity and Inequalities

G. D. Anderson·M. K. Vamanamurthy·M. Vuorinen
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Abstract

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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G. D. Anderson, M. K. Vamanamurthy, M. Vuorinen. 2007-01-09. Generalized Convexity and Inequalities. https://doi.org/10.1016/j.jmaa.2007.02.016

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