A refinement of the Schwarz-Pick estimates and the Carath éodory metric in several complex variables

AbstractIn this article, we first establish an asymptotically sharp result on the higher order Fr échet derivatives for bounded holomorphic mappings\(f(x) = f(0) + \sum\limits_{s = 1}^\infty {{{{D^{sk}}f(0)({x^{sk}})} \over {(sk)!}}} :\,{B_X} \to {B_Y}\), whereBX is the unit ball ofX. We next give a sharp result on the first order Fr échet derivative for bounded holomorphic mappings\(f(x) = f(0) + \sum\limits_{s = k}^\infty {{{{D^s}f(0)({x^s})} \over {s!}}} :\,{B_X} \to {B_Y}\), whereBX is the unit ball ofX. The results that we derive include some results in several complex variables, and extend the classical result in one complex variable to several complex variables.

Source: European Journal of Applied Physiology - April 16, 2024 Category: Physiology Source Type: research