Some bounds on convex mappings in several complex variables
Some bounds on convex mappings in several complex variables
复制标题
几个复变量中凸映射的一些界限
DOI：
10.2140/pjm.1994.165.295
复制
发表时间：
1994
影响因子：
0.6
通讯作者：
C. Thomas
中科院分区：
文献类型：
--
作者：
C. FitzGerald;C. Thomas
一个复变量中凸函数的系数界限以及增长和畸变定理可推广到多个变量。研究的全纯映射是在单位球或前三种经典类型之一的其他域中定义的。每个映射都以一对一的方式将其域置于凸集上。每个映射的坐标函数具有关于原点的多变量幂级数。对于这些幂级数的系数的某些组合，找到了最佳可能的上限。如果域是平面中的单位圆盘，则这些界限简化为凸函数的经典系数估计。作为一种应用，这些系数界限用于根据自变量的大小来获得每个映射的大小增长的最佳可能上限和下限。此外，还找到了对每个映射的各种导数的大小的估计。
The coefficient bounds and the Growth and Distortion Theorems for convex functions in one complex variable are generalized to several variables. The holomorphic mappings studied are defined in the unit ball or some other domain of one of the first three classical types. Each mapping takes its domain onto a convex set in a one-to-one fashion. The coordinate functions of each mapping have multivariable power series about the origin. The best possible upper bounds are found for certain combinations of the coefficients of these power series. In case the domain is the unit disk in the plane, these bounds reduce to the classical coefficient estimates for convex functions. As an application, these coefficient bounds are used to obtain the best possible upper and lower bounds on the growth of the magnitude of each mapping in terms of the magnitude of the independent variable. Also, estimates on the magnitudes of various derivatives of each mapping are found.