n-调和映射理论中的若干问题

The main aim of the project is to study the following three aspects. (1) Harmonic mappings: By using the theory of analytic functions and harmonic mappings, we plan to study harmonic Bloch mappings, and therefore generalize the corresponding results of Cima and Wogen which were published in Michigan Math. J.. Moreover, we also plan to discuss harmonic mappings whose real part of the derivative of the analytic part is large than the module of the derivative of the anti-analytic part, and therefore generalize the corresponding known results. (2) Biharmonic mappings: By using the theory of univalent functions and univalent harmonic mappings, we plan to study univalent biharmonic mappings with integer coefficients or half-integer coefficients, and therefore generalize the corresponding results of Friedman, Hiranuma and Sugawa which were published in Duke Math. J. and Comput. Mathods Funct. Theory, respectively; Moreover, we plan to make further discuss whether the univalent harmonic or biharmonic mappings with coefficients in uniformly discrete set of real numbers is finite. (3) n-harmonic mappings: We plan to discuss the geometric properties of n-harmonic mappings, and then establish relatively complete theory of n-harmonic mappings; Finally, we will investigate the properties of hyperbolic-harmonic mappings.This research has the important theoretical significance.

本项目研究以下三个方面的内容：（一）调和映射：利用解析函数论和调和映射理论研究调和Bloch映射，所得结果将推广Cima和Wogen发表在《Michigan Math. J.》的相应结果；此外，我们将讨论解析部分导数的实部大于反解析部分导数的模的调和映射，从而推广已有相关结论。（二）双调和映射：利用单叶函数论和单叶调和映射理论研究具有整数系数和半整数系数的单叶双调和映射，所得结果将推广和改进Friedman，Hiranuma和Sugawa分别发表在《Duke Math. J.》和《Comput. Mathods Funct. Theory》的结果，且我们将进一步讨论系数属于均匀离散数集的单叶调和映射和单叶双调和映射是否有限。（三）n-调和映射和双曲调和映射：主要研究n-调和映射的几何性质，从而建立较完善的n-调和映射理论；最后，讨论双曲调和映射的性质。此研究具有重要的理论意义。

本项目研究调和映射、双调和映射以及解析函数的性质，在研究工作中取得了较好的成绩，主要的工作如下：（一）调和映射和双调和映射：研究了调和Bloch映射的性质，解析部分导数的实部大于反解析部分导数的模的调和映射的性质，以及一致局部单叶调和映射的性质；确定了具有整数系数和半整数系数的保向单叶调和映射，具有复二次域整数系数的保向单叶的调和映射和双调和映射。（二）解析函数：作为导数实部大于零的解析函数类的推广，讨论了一类接近凸解析函数的性质；研究了$\alpha$ 阶凸函数的广义Zalcman猜测以及一类解析函数的一阶微分从属。