DISTORTION THEOREMS OF STARLIKE MAPPINGS IN SEVERAL COMPLEX VARIABLES

Abstract

In this paper, we first obtain a distortion theorem of Jacobian matrix $J_f(z)$ for biholomorphic starlike mapping $f$ along a unit direction on the unit polydisc. This result extends the classical distortion theorem of starlike functions to higher dimensions. We then give an upper bound estimate of distortion theorem for biholomorphic starlike mappings along a unit direction in a complex Banach space. Finally, we propose two conjectures on starlike mappings in several complex variables.