The authors present sharp transformation inequalities for the zero-balanced hypergeometric function \({}_2F_1(a,b;a+b;r)\) created by the transformations \(r\mapsto x=[(1-r)/(1+2r)]^3\) and \(r\mapsto 1-x\), \(r\mapsto u=(1/2)r(3+r)^2(1+r)^{-3}\) and \(r\mapsto 1-u\), \(r\mapsto p=(27/2)r(1+r)^4(1+4r+r^2)^{-3}\) and \(r\mapsto 1-p\), by showing the monotonicity properties of certain combinations in terms of hypergeometric functions and elementary functions, thus extending the transformation identities satisfied by \({}_2F_1(1/3,2/3;1;r)\) 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 \({}_2F_1(a,b;c;r)\) are obtained, too.

作者提出尖锐变换不等式零平衡几何函数\（{} _ 2F_1（A，B; A + B; R）\）由变换创建\（R \ mapsto X = [（1-R）/（ 1 + 2r）] ^ 3 \）和\（r \ mapsto 1-x \），\（r \ mapsto u =（1/2）r（3 + r）^ 2（1 + r）^ {-3 } \）和\（r \ mapsto 1-u \），\（r \ mapsto p =（27/2）r（1 + r）^ 4（1 + 4r + r ^ 2）^ {-3} \ ）和\（r \ mapsto 1-p \），通过显示某些组合的超几何函数和基本函数的单调性，从而扩展了\（{} _ 2F_1（1 / 3,2 / 3） ; 1; r）\）通过这三对转换，从根本上改善了相关的已知结果。有了这些结果，就可以获得广义Grötzsch环函数和出现在Ramanujan模块化方程式中的模块化函数的某些属性。也可以获得\（{} _ 2F_1（a，b; c; r）\）的其他一些属性。