Sharp Landen transformation inequalities for hypergeometric functions, with applications

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:474

Article

Qiu, Song-Liang¹  Ma, Xiao-Yan¹  Chu, Yu-Ming² 

[1] Zhejiang Sci Tech Univ, Dept Math, Hangzhou 310018, Zhejiang, Peoples R China

[2] Huzhou Univ, Dept Math, Huzhou 313000, Peoples R China

关键词: Hypergeometric function;    Landen transformation;    Generalized Grotzsch ring function;    Modular function;    Monotonicity;    Functional inequality;   

DOI  :  10.1016/j.jmaa.2019.02.018

来源: Elsevier

【 摘 要 】

The authors present sharp Landen transformation inequalities for the hypergeometric functions F-2(1)(a, b; a + b; x) and F-2(1)(a, b; (a + b + 1)/2; x), by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear (or rational) functions, thus giving complete solutions of the problem on extending the well-known Landen transformation identities for the complete elliptic integrals of the first kind to these two hyper-geometric functions, and substantially improving the related known results. As applications of these results, sharp Landen transformation inequalities are obtained for the generalized Grotzsch ring functions and the modular functions, which appear in Ramanujan's modular equations. Some other properties of hypergeometric functions and several properties of the Ramanujan constant are obtained, too. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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