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Parabolic starlike mappings of the unit ball B^n

Journal: Sahand Communications in Mathematical Analysis (Vol.3, No. 1)

Publication Date:

Authors : ;

Page : 63-70

Keywords : Roper-Suffridge extention operator; Biholomorphic mapping; Parabolic starlike function;

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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 $$Phi_{n,gamma}(f)(z)=left(f(z_1),left(f'(z_1)right)^gammahat{z}right),$$ where $gammain[0,1/2]$, $z=(z_1,hat{z})in B^n$ and $$Psi_{n,beta}(f)(z)=left(f(z_1),left(frac{f(z_1)}{z_1}right)^betahat{z}right),$$ 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 $B^n$.

Last modified: 2016-03-06 17:32:12