单复变几何函数论中若干经典结论在多复变中的推广

Geometric function theory in several complex variables orignates from function theory in one complex variable, which is abundant in research contents，and it is an important branch of function theory in several complex variables. Moreover, function theory in several complex variables belongs to international mathematical mainstream nowadays. After professors S. Gong and C.H.Fitzgerald etc first made breakthrough progress in geometric function theory in several complex variables in 1988, the research field is always active, and the results which attract people's attention are obtained, especially, where the results of chinese reseachers in the above field are the most prominent results. Based on research work over many years, we shall establish the sharp estimates of homogeneous expansions of each item for biholomorphic mappings （include quasi-convex mappings）with restricted conditions, establish the sharp estimates of main coefficients of each item for starlike mappings and quasi-convex mappings, establish the sharp distortion theorems for biholomorphic mappings （include quasi-convex mappings）with restricted conditions, and establish the Bohr's theorem between different dimmensional spaces （do not depend on automorphism） by applying the concept and method of modern mathematics (such as function theory in several complex variables, functional analysis and matrix theory). It is possible for our project team to make breakthrough progress with them due to our sufficient previous work.

多复变几何函数论源于单复变几何函数论，研究内容相当丰富，它是多复变函数论的重要分支，而多复变函数论又属当今国际数学的主流方向。自1988年龚升教授与C.H.Fitzgerald教授等人率先在多复变几何函数论领域取得突破性进展后，该研究领域一直十分活跃，取得了许多令人瞩目的成果，尤其以国内学者的成果最为突出。在多年研究工作的基础上，本项目拟用现代数学（诸如多复变、泛函分析、矩阵论等）的概念和方法，建立限制条件下双全纯映射(含准凸映射)齐次展开式各项的精确估计；建立星形映射与准凸映射主要系数各项的精确估计； 建立限制条件下双全纯映射(含准凸映射)精确的偏差定理；建立不同维数空间的Bohr 定理（不依赖自同构）。项目组前期准备工作较充分，有望取得突破性进展。

多复变几何函数论是多复变函数论的重要分支，而多复变函数论又属当今国际数学的主流方向。本项目以四个在单复变几何函数论中结论优美而深刻，又可望在多复变成立的问题作为研究对象，取得的五个主要结果如下：(1) 建立了多复变数限制条件下全纯映射缺项系数范数型和泛函型Bohr不等式，深化了多复变Bohr定理的研究；(2) 利用Carathéodory 度量给出关于自映射的齐次展开式估计式，从而证明多复变数不同维数之间全纯映射精细的Bohr定理成立，为研究多复变Bohr定理提供新工具；(3) 建立了关于三类星形映射子族Jacobi 行列式型偏差定理，为多复变重要问题星型映射Jacobi 行列式型偏差定理的研究作了有益探索；(4)给出高维Reinhardt域上星形映射子族、准凸映射子族齐次展开式中主要系数的精确估计，为多复变Bieberbach猜想研究提供了新思路；(5) 建立了限制条件下多复变双全纯映射三种类型偏差定理、全部项齐次展开式的精确估计与Feket Szego不等式，有助于对限制条件下多复变双全纯映射的几何性质有了新的认识。Bohr半径，星形映射偏差定理和多复变数Bieberbach猜想属多复变几何函数论研究领域的重要问题，对这些问题的研究， 有助于深化和丰富多复变几何函数论的研究成果，从而进一步提升项目组的研究水平。