Abstract and Applied Analysis
Volume 2012 (2012), Article ID 932061, 11 pages
http://dx.doi.org/10.1155/2012/932061
Research Article

Landen Inequalities for Zero-Balanced Hypergeometric Functions

Slavko Simić1 and Matti Vuorinen2

1Mathematical Institute, SANU, Kneza Mihaila 36, 11000 Belgrade, Serbia
2Department of Mathematics, University of Turku, 20014 Turku, Finland

Received 22 October 2011; Accepted 6 January 2012

Academic Editor: Paul Eloe

Copyright © 2012 Slavko Simić and Matti Vuorinen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For zero-balanced Gaussian hypergeometric functions 𝐹 ( 𝑎 , 𝑏 ; 𝑎 + 𝑏 ; 𝑥 ) , 𝑎 , 𝑏 > 0 , we determine maximal regions of 𝑎 𝑏 plane where well-known Landen identities for the complete elliptic integral of the first kind turn on respective inequalities valid for each 𝑥 ∈ ( 0 , 1 ) . Thereby an exhausting answer is given to the open problem from the work by Anderson et al., 1990.