Generalized convexity and inequalities

: Anderson G., Vamanamurthy M., Vuorinen M.

: 2007

: Journal of Mathematical Analysis and Applications

: 335

: 2

: 1294

: 1308

: 15

: 0022-247X

DOI: https://doi.org/10.1016/j.jmaa.2007.02.016

: http://api.elsevier.com/content/abstract/scopus_id:34447529172

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 m, m ∈ M, we say that a function f : R → R is (m, m)-convex if f (m (x, y)) ≤ m (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 (m, m)-convexity on m and m and give sufficient conditions for (m, m)-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.