کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
4616635 | 1339355 | 2013 | 19 صفحه PDF | دانلود رایگان |
![عکس صفحه اول مقاله: Log-convexity and log-concavity for series in gamma ratios and applications Log-convexity and log-concavity for series in gamma ratios and applications](/preview/png/4616635.png)
Polynomial sequence {Pm}m≥0{Pm}m≥0 is qq-logarithmically concave if Pm2−Pm+1Pm−1 is a polynomial with nonnegative coefficients for any m≥1m≥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 qq-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 Kummer function, the ratio of the Gauss functions and the generalized hypergeometric function obtained as direct applications of the general theorems.
Journal: Journal of Mathematical Analysis and Applications - Volume 406, Issue 2, 15 October 2013, Pages 400–418