Parabolic starlike mappings of the unit ball $B^n$

Abstract

Let $f$ be a locally univalent function on the unit disk $U$. We consider the normalized extensions of $f$ to the Euclidean unit ball $B^nsubseteqmathbb{C}^n$ given by $$Phi_{n,gamma}(f)(z)=left(f(z_1),(f'(z_1))^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),(frac{f(z_1)}{z_1})^betahat{z}right),$$ in which $betain[0,1]$, $f(z_1)neq 0$ and $z=(z_1,hat{z})in B^n$. In the case $gamma=1/2$, the function $Phi_{n,gamma}(f)$ reduces to the well known Roper-Suffridge extension operator. By using different methods, we prove that if $f$ is parabolic starlike mapping on $U$ then $Phi_{n,gamma}(f)$ and $Psi_{n,beta}(f)$ are parabolic starlike mappings on $B^n$.