JANOWSKI STARLIKENESS IN SEVERAL COMPLEX VARIABLES AND COMPLEX HILBERT SPACES

Abstract

In this paper, we consider two new subclasses, $S^*(a,b,B^n)$ and ${\mathcal A}S^*(a,b,B^n)$, of the class of starlike mappings on $B^n$ ($a,b\in \mathbb{R}$, $|a-1|\lt b\le a$, and $B^n$ is the Euclidean unit ball in $\mathbb{C}^n$). The class $S^*(a,b,B)$ is the $n$-dimensional version of Janowski class of one variable starlike functions. We obtain sharp growth results and upper distortion estimates for these two classes of starlike mappings. We also derive sufficient conditions for normalized holomorphic mappings (expressed in terms of their coefficient bounds) to belong to one of the classes $S^*(a,b,B^n)$, respectively ${\mathcal A}S^*(a,b,B^n)$. Finally, similar notions on the unit ball in a complex Hilbert space are analogously presented.