交错代数上slice正则函数的若干几何函数论问题研究

The slice regular function over quaternions is the extension of complex analysis in the non-commutative algebras, and is one of the hot spots in the study of hypercomplex analysis in the last decade. Nowdays, slice regular functions have been widely used in the quaternionic theories of operators, functional analysis and Schur analysis. However, there are many gaps in the corressponding geometric function theory. Therefore, this project will study the geometric function theory of slice regular functions over alternative algebras (e.g., quaternions, octonions) in which we mainly explore the coefficient estimate (e.g., Bieberbach conjecture, Fekete-Szegö inequality), proper mapping, Lu Qikeng problem, Bloch-Landau theorem, growth and distortion theorems. Specially, we hope to overcome the non-associativity and then extend the non-commutative technique of processing quaternionic slice starlike functions and convex functions to the octonionic setting. At the same time, this project will establish the generalized form of Bernstein inequality for slice regular polynomials and give its applications in approximation theory. Finally, we shall discuss whether the classical Gauss-Lucas theorem and Turan inequality are valid for slice regular polynomials. The study of the above problems would help to reveal the intrinsic relations and fundamental differences between holomorphic functions and slice regular functions.

四元数slice正则函数是复分析在非交换代数上的推广，是近十年超复分析的研究热点之一。slice正则函数在四元数算子理论、泛函分析、Schur分析中取得了极为广泛的应用，然其几何函数论方面还有许多空白。因此本项目将研究交错代数（如四元数、八元数）上的slice正则函数的几何函数论，重点探究其系数估计（如Bieberbach猜测、Fekete-Szegö不等式）、逆紧映射、陆启铿问题、Bloch-Landau定理、增长和偏差定理。其中，希望克服非结合性，将我们处理四元数slice星形函数、凸函数的非交换技巧推广到八元数上。同时，本项目将研究slice正则多项式的Bernstein不等式的推广形式，并给出其在逼近论中的应用；探讨经典的Gauss-Lucas定理、Turan不等式对于slice正则多项式是否成立。上述问题的研究将有助于揭示全纯函数与slice正则函数的内在联系和根本区别。

多复变函数论是现代数学的主流方向之一，几何函数论是其重要的组成部分。切片正则函数理论是复分析在非交换代数上的高维推广，在近十五年里得到了充分的发展，而且该理论可以广泛应用于四元数算子理论、逼近论、twistor几何等诸多领域。本项目以切片正则函数中几何函数论为出发点，主要研究了逆紧映射、Bohr半径及推广形式、Bernstein不等式的L^p形式、陆启铿问题、不确定性原理的算子形式。在项目的资助下，共发表学术论文8篇，如Proceedings of the Royal Society of Edinburgh、Annali di Matematica Pura ed Applicata、Annales Fennici Mathematici、Complex Analysis and Operator Theory、中国科学：数学。这些问题的研究，不但丰富了复分析和非交换代数的基本理论，而且促进了学科间的交叉发展。在项目组成员的共同努力下，圆满完成了项目中的预期目标。