AN EXTENSION OPERATOR ASSOCIATED WITH CERTAIN $G$-LOEWNER CHAINS

Abstract

In this paper we are concerned with an extension operator $\Phi_{n,\alpha,\beta}$ that provides a way of extending a locally univalent function $f$ on the unit disc $U$ to a locally biholomorphic mapping $F\in H(B^{n})$. By using the method of Loewner chains, we prove that if $f$ can be embedded as the first element of a $g$-Loewner chain on the unit disc, where $g(\zeta)=\frac{1-\zeta}{1+(1-2\gamma)\zeta}$ for $|\zeta|\lt 1$ and $\gamma \in (0,1)$, then $F=\Phi_{n,\alpha,\beta}(f)$ can also be embedded as the first element of a $g$-Loewner chain on $B^n$, whenever $\alpha\in [0,1]$, $\beta \in [0,1/2]$, $\alpha +\beta \leq 1$. In particular, if $f$ is starlike of order $\gamma \in (0,1)$ on $U$, then $F=\Phi_{n,\alpha,\beta}(f)$ is also starlike of order $\gamma$ on $B^n$. Also, if $f$ is spirallike of type $\delta$ and order $\gamma$ on $U$, where $\delta\in (-\pi/2,\pi/2)$ and $\gamma \in (0,1)$, then $F=\Phi_{n,\alpha,\beta}(f)$ is spirallike of type $\delta$ and order $\gamma$ on $B^n$. We also obtain a subordination preserving result under the operator $\Phi_{n,\alpha,\beta}$ and we consider some radius problems associated with this operator.