Distortion theorems, Lipschitz continuity and their applications for Bloch type mappings on bounded symmetric domains in C^n

Abstract

Let BX be a bounded symmetric domain realized as the unit ball of an ndimensional JB∗-triple X = (C, ‖ · ‖X). In this paper, we give a new definition of Bloch type mappings on BX and give distortion theorems for Bloch type mappings on BX . When BX is the Euclidean unit ball in C, this new definition coincides with that given by Chen and Kalaj or by the author. As a corollary of the distortion theorem, we obtain the lower estimate for the radius of the largest schlicht ball in the image of f centered at f(0) for α-Bloch mappings f on BX . Next, as another corollary of the distortion theorem, we show the Lipschitz continuity of (detB(z, z))1/2n| detDf(z)|1/n for Bloch type mappings f on BX with respect to the Kobayashi metric, where B(z, z) is the Bergman operator on X , and use it to give a sufficient condition for the composition operator Cφ to be bounded from below on the Bloch type space on BX , where φ is a holomorphic self mapping of BX . In the case BX = B , we also give a necessary condition for Cφ to be bounded from below which is a converse to the above result. Finally, as another application of the Lipschitz continuity, we obtain a result related to the interpolating sequences for the Bloch type space on BX .