Taiwanese Journal of Mathematics

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

Yu-Can Zhu and Ming-Sheng Liu

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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.

Article information

Source
Taiwanese J. Math., Volume 14, Number 2 (2010), 359-372.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500405794

Digital Object Identifier
doi:10.11650/twjm/1500405794

Mathematical Reviews number (MathSciNet)
MR2655774

Zentralblatt MATH identifier
1200.32013

Subjects
Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)

Keywords
roper-Suffridge extension operator biholomorphic starlike mapping almost spirallike mapping of type $\beta$ and order $\alpha$ spirallike mapping of type $\beta$ and order $\alpha$

Citation

Zhu, Yu-Can; Liu, Ming-Sheng. THE GENERALIZED ROPER-SUFFRIDGE EXTENSION OPERATOR ON REINHARDT DOMAIN $D_p$. Taiwanese J. Math. 14 (2010), no. 2, 359--372. doi:10.11650/twjm/1500405794. https://projecteuclid.org/euclid.twjm/1500405794


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