A1 Refereed original research article in a scientific journal
Generalized convexity and inequalities
Authors: Anderson G., Vamanamurthy M., Vuorinen M.
Publication year: 2007
Journal: Journal of Mathematical Analysis and Applications
Journal name in source: Journal of Mathematical Analysis and Applications
Volume: 335
Issue: 2
First page : 1294
Last page: 1308
Number of pages: 15
ISSN: 0022-247X
DOI: https://doi.org/10.1016/j.jmaa.2007.02.016
Web address : http://api.elsevier.com/content/abstract/scopus_id:34447529172
Abstract
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.
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.
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