Sharp Landen transformation inequalities for hypergeometric functions, with applications
By Song-Liang Qiu, Xiao-Yan Ma, Y. Chu. Journal of Mathematical Analysis and Applications, 2019
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Abstract The authors present sharp Landen transformation inequalities for the hypergeometric functions F 1 2 ( a , b ; a + b ; x ) and F 1 2 ( 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 hypergeometric 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.
Subfields: Algebra and Number Theory, Applied Mathematics
View Paper: https://www.semanticscholar.org/paper/3c2e07ff929f9c3cb191735c6299bfcdeacfd7a8
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In Algebra and Number Theory
In Applied Mathematics
Citations in the corpus
- Sharp bounds for the Toader mean of order 3 in terms of arithmetic, quadratic and contraharmonic means (2020, background): “is the complete elliptic integral of the second kind [12,18,30,38,44,50,52] and it is the special case of the Gaussian hypergeometric function [31,41,42,49,57]”