Log-convexity and log-concavity for series in gamma ratios and applications

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:406

Article

Karp, D. B.¹ 

[1] Far Eastern Fed Univ, Vladivostok 690950, Russia

关键词: Gamma function;    Log-concavity;    Log-convexity;    q-log-concavity;    Wright log-concavity;    Turan inequality;    Kummer function;    Generalized hypergeometric function;   

DOI  :  10.1016/j.jmaa.2013.04.061

来源: Elsevier

【 摘 要 】

Polynomial sequence {Pm}(m >= 0), is q-logarithmically concave if p(m)(2)-Pm+1 P(m-1)i is a polynomial with nonnegative coefficients for any m >= 1. We introduce an analogue of this notion for formal power series whose coefficients are nonnegative continuous functions of a parameter. Four types of such power series are considered where the parameter dependence is expressed by a ratio of gamma functions. We prove six theorems stating various forms of q-logarithmic concavity and convexity of these series. The main motivating examples for these investigations are hypergeometric functions. In the last section of the paper we present new inequalities for the Mummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems. (C) 2013 Elsevier Inc. All rights reserved.

【 授权许可】

Free