THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR ON REINHARDT DOMAIN $D_p$

Abstract

We define the generalized Roper-Suffridge extension operator $\Phi_{n, \beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)$ on Reinhardt domain $D_p$ as \[ \Phi_{n, \beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)(z) = \bigg(f(z_1), \bigg(\frac{f(z_1)}{z_1}\bigg)^{\beta_2} \Big(f'(z_1)\Big)^{\gamma_2} z_2, \cdots, \bigg(\frac{f(z_1)}{z_1}\bigg)^{\beta_n} \Big(f'(z_1)\Big)^{\gamma_n} z_n \bigg) \] for $z = (z_1, z_2, \cdots, z_n) \in D_p$, where $D_p = \Big\{(z_1, z_2, \cdots, z_n) \in C^n: \sum\limits_{j=1}^{n} |z_j|^{p_j} \lt 1 \Big\}$, $p = (p_1, p_2, \cdots, p_n)$, $p_j \gt 0$, $0 \leq \gamma_j \leq 1 - \beta_j$, $0 \leq \beta_j \leq 1$, $j = 1,2,\cdots,n$, and we choose the branch of the power functions such that $\Big(\frac{f(z_1)}{z_1}\Big)^{\beta_j}|_{z_1=0} = 1$ and $(f'(z_1))^{\gamma_j}|_{z_1=0}=1$, $j = 2,\cdots,n$. In the present paper, we show that the operator $\Phi_{n,\beta_2, \gamma_2, \cdots, \beta_n, \gamma_n}(f)$ preserves almost spirallike mapping of type $\beta$ and order $\alpha$ and spirallike mapping of type $\beta$ and order $\alpha$ on $D_p$ for some suitable constants $\beta_j, \gamma_j, p_j$. The results improve the corresponding results of earlier authors.