## Taiwanese Journal of Mathematics

### 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.

#### Article information

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

Dates
First available in Project Euclid: 18 July 2017

https://projecteuclid.org/euclid.twjm/1500405794

Digital Object Identifier
doi:10.11650/twjm/1500405794

Mathematical Reviews number (MathSciNet)
MR2655774

Zentralblatt MATH identifier
1200.32013

#### 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

#### References

• I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker Inc, New York, 2003.
• I. Graham and G. Kohr, Univalent mappings associated with the Roper-Suffridge extension operator, J. Anal. Math., 81 (2000), 331-342.
• I. Graham, Growth and covering theorems associated with the Roper-Suffridge extension operator, Proc. Amer. Math. Soc., 127 (1999), 3215-3220.
• I. Graham, H. Hamada and G. Kohr, et al. Extension operators for locally univalent mappings, Michigan Math. J., 50 (2002), 37-55.
• I. Graham, H. Hamada and G. Kohr, Parametric representation of univalent mappings in several complex variables, Canad. J. Math., 54 (2002), 324-351.
• I. Graham, G. Kohr and M. Kohr, Loewner chains and Roper-Suffridge extension operator, J. Math. Anal. Appl., 247 (2000), 448-465.
• I. Graham and G. Kohr, An extension theorem and subclass of univalent mappings in several complex variables, Complex Variables, 47(1), (2002), 59-72.
• I. Graham and G. Kohr, The Roper-Suffridge extension operator and classes of biholomorphic mappings, Science in China, Series A, 49(11) (2006), 1539-1552.
• S. Gong, Convex and Starlike Mappings in Several Complex Variables, New York, Kluwer Academic Publishers, 1998.
• S. Gong and T. S. Liu, On Roper-Suffridge extension operator, J. Anal. Math., 88 (2002), 397-404.
• S. Gong and T. S. Liu, The generalized Roper-Suffridge extension operator, J. Math. Anal. Appl., 284 (2003), 425-434.
• M. S. Liu and Y. C. Zhu, On the generalized Roper-Suffridge extension operator in Banach spaces, International J. Math. Math. Sci., 2005(8) (2005), 1171-1187.
• M. S. Liu and Y. C. Zhu, On $\varepsilon$ quasi-convex mappings in the unit ball of a complex Banach space, J. Math. Anal. Appl., 323 (2006), 1047-1070.
• M. S. Liu and Y. C. Zhu, The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains, Science in China, Series A, 37(10) (2007), 1193-1206, (in Chinese).
• X. S. Liu, The generalized Roper-Suffridge extension operator for some biholomorphic mappings, J. Math. Anal. Appl., 324(1) (2006), 604-614.
• X. S. Liu and T. S. Liu, The generalized Roper-Suffridge extension operator on a Reinhardt domain and the unit ball in a complex Hilbert space, Chin. Ann. of Math. Ser. A, 26(5) (2005), 721-730, (in Chinese).
• X. S. Liu and T. S. Liu, The generalized Roper-Suffridge extension operator for spirallike mapping of type $\beta$ and order $\alpha$, Chin. Ann. Math. Ser. A, 27(6) (2006), 789-798, (in Chinese).
• J. A. Pfaltzgraff and T. J. Suffridge, Close-to-starlike holomorphic functions of several variables, Pacific. J. Math., 57 (1975), 271-279.
• K. Roper and T. J. Suffridge, Convex mappings on the unit ball of $C^n$, J. Anal. Math., 65 (1995), 333-347.
• T. J. Suffridge, Starlikeness, convexity and other geometric properties of holomorphic maps in higher dimensions, Lecture Notes in Math., 599 (1976), 146-159.
• W. J. Zhang and T. S. Liu, On decomposition theorem of normalized biholomorphic convex mappings in Reinhardt domains, Science in China, Series A, 46(1) (2003), 94-106.
• Y. C. Zhu and M. S. Liu, The generalized Roper-Suffridge extension operator in Banach spaces (I), Acta Mathematica Sinica, Chinese Series, (in Chinese), 50(1) (2007), 189-196.
• Y. C. Zhu and M. S. Liu, The generalized Roper-Suffridge extension operator in Banach spaces (II), J. Math. Anal. Appl., 303(2) (2005), 530-544.
• Y. C. Zhu and M. S. Liu, The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains, Science in China, Series A, 50(12) (2007), 1781-1794.