Transformation Properties of Hypergeometric Functions and Their Applications

The authors present sharp transformation inequalities for the zero-balanced hypergeometric function 2F1(a,b;a+b;r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2F_1(a,b;a+b;r)$$\end{document} created by the transformations r↦x=[(1-r)/(1+2r)]3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto x=[(1-r)/(1+2r)]^3$$\end{document} and r↦1-x\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto 1-x$$\end{document}, r↦u=(1/2)r(3+r)2(1+r)-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto u=(1/2)r(3+r)^2(1+r)^{-3}$$\end{document} and r↦1-u\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto 1-u$$\end{document}, r↦p=(27/2)r(1+r)4(1+4r+r2)-3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto p=(27/2)r(1+r)^4(1+4r+r^2)^{-3}$$\end{document} and r↦1-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r\mapsto 1-p$$\end{document}, by showing the monotonicity properties of certain combinations in terms of hypergeometric functions and elementary functions, thus extending the transformation identities satisfied by 2F1(1/3,2/3;1;r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2F_1(1/3,2/3;1;r)$$\end{document} with these three pairs of transformations and substantively improving the related known results. With these results, some properties are obtained for the generalized Grötzsch ring functions and the modular functions appearing in Ramanujan’s modular equations. Some other properties of 2F1(a,b;c;r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}_2F_1(a,b;c;r)$$\end{document} are obtained, too.