^{}Department of Mathematics, Faculty of Basic Science, University of Bonab, P.O. Box 5551-761167, Bonab, Iran.

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 $B^n\subseteq\mathbb{C}^n$ given by $$\Phi_{n,\gamma}(f)(z)=\left(f(z_1),(f'(z_1))^\gamma\hat{z}\right),$$ where $\gamma\in[0,1/2]$, $z=(z_1,\hat{z})\in B^n$ and $$\Psi_{n,\beta}(f)(z)=\left(f(z_1),(\frac{f(z_1)}{z_1})^\beta\hat{z}\right),$$ in which $\beta\in[0,1]$, $f(z_1)\neq 0$ 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 $B^n$.

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