关键词: Biholomorphic mappings, starlike mappings, subordination, Loewner chain

Abstract: This note is devoted to applying the principle of subordination in order to explore the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator with special analytic properties. First, we prove that both the Roper-Suffridge extension operator and the Pfaltzgraff-Suffridge extension operator preserve subordination. As applications, we obtain that if $\beta\in[0,1],\gamma\in[0,\frac{1}{r}]$ and $\beta+\gamma\leq1$, then the Roper-Suffridge extension operator $$ \Phi_{\beta,\,\gamma}(f)(z)=\left(f(z_{1}), \left(\frac{f(z_1)}{z_1}\right)^{\beta}(f'(z_{1}))^{\gamma}w\right),\,\,z\in \Omega_{p,r} $$ preserves an almost starlike mapping of complex order $\lambda$ on $\Omega_{p,r}=\{z=(z_1,w)\in \mathbb C\times X :|z_1|^{p}+\|w\|_X^{r}<1\}$, where $1\leq p\leq 2$, $r\geq 1$ and $X$ is a complex Banach space. Second, by applying the principle of subordination, we will prove that the Pfaltzgraff-Suffridge extension operator preserves an almost starlike mapping of complex order $\lambda$. Finally, we will obtain the lower bound of distortion theorems associated with the Roper-Suffridge extension operator. This subordination principle seems to be a new idea for dealing with the Loewner chain associated with the Roper-Suffridge extension operator, and enables us to generalize many known results from $p=2$ to $1\leq p\leq 2$.

Key words: Biholomorphic mappings, starlike mappings, subordination, Loewner chain