第一类典型域上的Bloch常数

Bloch Constant on Classical Domain of the First Type

引入第一类典型域R_I(m,n)上的全纯映照子族H_k(R_I(m,n)),当k→+∞时,该映照族就是R_I(m,n)上的局部双全纯映照族.建立了H_k(R_I(m,n))上的Bonk偏差定理.当k=1和k→+∞时,其结果分别都回到了FitzGerad和龚升关于典型域R_I(m,n)上的Bonk偏差定理.当m=n=1时,其结果又回到了Liu和Minda在单位圆盘上的偏差定理.应用偏差定理,给出了映照族H_k(R_I(m,n))上的Bloch常数估计,其结果补全了从k=1和k→+∞之间的R_I(m,n)上Bloch常数估计的所有结果,而且把单位球上的Bloch常数估计推广到R_I(m,n)上.

In this paper,various subfamilies Hk（RI（m,n）） of holomorphic mappings denned in the first classical domain RI（m,n） are introduced.When fc tends to ＋∞, this family reduces to the class of locally biholomorphic mappings on RI（m,n）.We establish the Bonk distortion theorems for Hk（RI（m,n））.In particular,when k = 1 and k→＋∞,the theorems reduce to that of Fitzgerald and Gong,respectively.When m = n = 1,this distortion theorem coincides with Liu and Minda as the unit disk case. As applications of the Bonk distortion theorems,various estimates of Bloch constants for these subfamilies of holomorphic mappings on RI（m,n） are obtained.We not only give all Bloch estimates of holomorphic mappings between 1 k ＋∞defined in RI（m,n）,but also extend our early Bloch constant estimates of the unit ball to the classical domain of the first type RI（m,n）.