本项目着重研究拟共形映射理论中的超几何函数的分析和几何性质及其在 Ramanujan 模方程理论中的应用，主要研究内容为以下三部分：（1） Teichmüller 环容量和 Robin 容量的精确或渐进精确不等式、Hübner 上界函数的无穷乘积表达式和 Hersch-Pfluger 偏差函数的精确估值，并由此改进拟共形 Schwarz 引理和 Ramanujan 模方程解的精确估计；（2）符号差分别为 1/4 和 1/6 的广义模方程及其相关的超几何函数的基本性质，给出超几何级数比值函数的无穷乘积表达式及其对应超几何函数二次变换的对偶形式；（3）广义 Grötzsch 环函数和广义 Hersch-Pfluger 偏差函数的分析性质，给出 Ramanujan 遗留手稿中若干至今尚未得到证明的有关模方程解的代数恒等式的严格证明。本项目学科交叉性强，应用前景广。

In this project, our research will focus on the analytic and geometric properties of hypergeometric functions in quasiconformal mappings, and their applications in Ramanujan's modular equations. The main contants include the following three parts: (1) The precise or asymptotical precise inequalities for the capacities of Teichmüller ring and Robin, infinity product formula for Hübner upper bound function and sharp evaluations of the Hersch-Pfluger distortion function. On this basic, we will improve the well known quasiconformal Schwarz lemma and the estimation for the solutions to Ramanujan's modular equations. (2) The basic properties of the generalized modular equations with signatures 1/4 and 1/6 together with their related hypergeometric functions, the infinity-product representations for the quotients of hypergeometric series and the dual form of the quadratic transformations for their corresponding hypergeometric functions. (3) The analytic properties of the generalized Grötzsch ring and generalized Hersch-Pfluger distortion functions, and the strict mathematical proofs for several algebraic identities involving the solutions to Ramanujan's modular equations which was appeared in the Ramanujan's lost notebooks and have not yet been proved. This project is interdisciplinary and has broad application prospects.