Log-convexity and log-concavity of hypergeometric-like functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:364

Article

Karp, D.¹  Sitnik, S. M.² 

[1] Russian Acad Sci, Inst Appl Math, Far Eastern Branch, Vladivostok 690022, Russia

[2] Voronezh Polytech Inst, Minist Internal Affairs Russian Federat, Voronezh, Russia

关键词: Hypergeometric function;    Kummer function;    Gauss function;    Log-convexity;    Log-concavity;    Turan inequality;   

DOI  :  10.1016/j.jmaa.2009.10.057

来源: Elsevier

【 摘 要 】

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a bar right arrow Sigma f(k)(a)(k)x(k), a bar right arrow Sigma f(k)Gamma(a + k)x(k) and a bar right arrow Sigma f(k)x(k)/(a)(k). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turan inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free