线性测度与K-拟共形调和映射

Linear measure and K-quasiconformal harmonic mappings

本文主要讨论调和映射与线性测度之间的关系.首先,证明单位圆周上任意可测集在单叶且保向的调和映射作用下的边界线性测度的最佳偏差定理.其次,建立K-拟共形调和映射的Schwarz型引理,并利用K-拟共形调和映射来刻画M-Lavrentiev域.最后,讨论具有有限径向长度的K-拟共形调和映射的系数估计以及径向长度与曲线周长之间的比值.

In this paper, we investigate the relationships between linear measure and harmonic mappings. Firstly, we obtain the sharp linear measure distortion theorem of the image of any measurable subset of the unit circle under sense-preserving and univalent harmonic mappings. Secondly, we establish a Schwarz type lemma of K-quasiconformal harmonic mappings, and give a characterization of M-Lavrentiev domains by using K-quasiconformal harmonic mappings. Finally, we discuss coefficient estimates on a class of K-quasiconformal harmonic mappings with the finite radial length.