Jian Fei WANG, Tai Shun LIU
In this paper, various subfamilies Hk(RI(m,n))of holomorphic mappings defined in the first classical domain RI(m,n)are introduced. When k 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).