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Extension Operators for Biholomorphic Mappings

Part of: Holomorphic mappings and correspondences Geometric function theory

Published online by Cambridge University Press:  07 January 2019

Jianfei Wang  and
Danli Zhang
Jianfei Wang
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China Email: wangjf@mail.ustc.edu.cn
Danli Zhang
Affiliation:
Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China Email: 465472962@qq.com
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Abstract

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Suppose that $D\subset \mathbb{C}$ is a simply connected subdomain containing the origin and $f(z_{1})$ is a normalized convex (resp., starlike) function on $D$. Let

$$\begin{eqnarray}\unicode[STIX]{x1D6FA}_{N}(D)=\bigg\{(z_{1},w_{1},\ldots ,w_{k})\in \mathbb{C}\times \mathbb{C}^{n_{1}}\times \cdots \times \mathbb{C}^{n_{k}}:\Vert w_{1}\Vert _{p_{1}}^{p_{1}}+\cdots +\Vert w_{k}\Vert _{p_{k}}^{p_{k}}<\frac{1}{\unicode[STIX]{x1D706}_{D}(z_{1})}\bigg\},\end{eqnarray}$$
where $p_{j}\geqslant 1$, $N=1+n_{1}+\cdots +n_{k}$, $w_{1}\in \mathbb{C}^{n_{1}},\ldots ,w_{k}\in \mathbb{C}^{n_{k}}$ and $\unicode[STIX]{x1D706}_{D}$ is the density of the hyperbolic metric on $D$. In this paper, we prove that
$$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)(z_{1},w_{1},\ldots ,w_{k})=(f(z_{1}),(f^{\prime }(z_{1}))^{1/p_{1}}w_{1},\ldots ,(f^{\prime }(z_{1}))^{1/p_{k}}w_{k})\end{eqnarray}$$
 is a normalized convex (resp., starlike) mapping on $\unicode[STIX]{x1D6FA}_{N}(D)$. If $D$ is the unit disk, then our result reduces to Gong and Liu via a new method. Moreover, we give a new operator for convex mapping construction on an unbounded domain in $\mathbb{C}^{2}$. Using a geometric approach, we prove that $\unicode[STIX]{x1D6F7}_{N,1/p_{1},\ldots ,1/p_{k}}(f)$ is a spiral-like mapping of type $\unicode[STIX]{x1D6FC}$ when $f$ is a spiral-like function of type $\unicode[STIX]{x1D6FC}$ on the unit disk.

Keywords

biholomorphic mapping𝜀-starlike mappingspiral-like mapping

MSC classification

Primary: 32H02: Holomorphic mappings, (holomorphic) embeddings and related questions
Secondary: 30C45: Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
Type
Article
Information
Canadian Mathematical Bulletin , Volume 62 , Issue 3 , September 2019 , pp. 671 - 679
Copyright
© Canadian Mathematical Society 2018 

Footnotes

This project was partially supported by the National Natural Science Foundation of China (Nos. 11671362, 11471111, & 11571105) and the Natural Science Foundation of Zhejiang Province (No. LY16A010004).

References

Beardon, A. F. and Minda, D., The hyperbolic metric and geometric function theory . In: Quasiconformal mappings and their applications , Narosa, New Delhi, 2007, pp. 9–56.Google Scholar
Elin, M., Extension operators via semigroups . J. Math. Anal. Appl. 377(2011), 239–250. https://doi.org/10.1016/j.jmaa.2010.10.025.Google Scholar
Elin, M. and Levenshtein, M., Covering results and perturbed Roper-Suffridge operators . Complex Anal. Oper. Theory. 8(2014), 25–36. https://doi.org/10.1007/s11785-012-0259-1.Google Scholar
Feng, S. and Liu, T., The generalized Roper-Suffridge extension operator . Acta Math. Sci. Ser. B 28(2008), 63–80. https://doi.org/10.1016/S0252-9602(08)60007-7.Google Scholar
Feng, S. and Yu, L., Modified Roper-Suffridge operator for some holomorphic mappings . Front. Math. China 6(2011), 411–426. https://doi.org/10.1007/s11464-011-0116-y.Google Scholar
Gong, S. and Liu, T., On Roper-Suffridge extension operator . J. Anal. Math. 88(2002), 397–404. https://doi.org/10.1007/BF02786583.Google Scholar
Gong, S. and Liu, T., The generalized Roper-Suffridge extension operator . J. Math. Anal. Appl. 284(2003), 425–434. https://doi.org/10.1016/S0022-247X(02)00400-6.Google Scholar
Graham, I., Hamada, H., Kohr, G., and Suffridge, T. J., Extension operators for locally univalent mappings . Michigan. Math. J. 50(2002), 37–55. https://doi.org/10.1307/mmj/1022636749.Google Scholar
Graham, I., Hamada, H., Kohr, G., and Kohr, M., Spirallike mappings and univalent subordination chains in ℂ n . Ann. Sc. Norm. Super. Pisa Cl. Sci. 7(2008), 717–740.Google Scholar
Graham, I. and Kohr, G., Univalent mappings associated with the Roper-Suffridge extension operator . J. Anal. Math. 81(2000), 331–342. https://doi.org/10.1007/BF02788995.Google Scholar
Liu, M. and Zhu, Y., On some sufficient conditions for starlikeness of order 𝛼 in ℂ n . Taiwanese J. Math. 10(2006), 1169–1182. https://doi.org/10.11650/twjm/1500557296.Google Scholar
Liu, X., The generalized Roper-Suffridge extension operator for some biholomorphic mappings . J. Math. Anal. Appl. 324(2006), 604–614. https://doi.org/10.1016/j.jmaa.2005.12.037.Google Scholar
Liu, T. and Xu, Q., Loewner chains associated with the generalized Roper-Suffridge extension operator . J. Math. Anal. Appl. 322(2006), 107–120. https://doi.org/10.1016/j.jmaa.2005.08.055.Google Scholar
Roper, K. and Suffridge, T. J., Convex mappings on the unit ball of ℂ n . J. Anal. Math. 65(1995), 333–347. https://doi.org/10.1007/BF02788776.Google Scholar
Wang, J., Modified Roper-Suffridge operator for some subclasses of starlike mappings on Reinhardt domains . Acta Math. Sci. Ser. B 33(2013), 1627–1638. https://doi.org/10.1016/S0252-9602(13)60110-1.Google Scholar
Wang, J. and Liu, T., The Roper-Suffridge extension operator and its applications to convex mappings in ℂ2 . Trans. Amer. Math. Soc. https://doi.org/10.1090/tran/7221.Google Scholar
Xu, Q., Liu, T., and Liu, X., On a subclass of close-to-convex mappings . Complex Anal. Oper. Theory. 9(2015), 275–286. https://doi.org/10.1007/s11785-014-0357-3.Google Scholar
Zhu, Y. and Liu, M., The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains . Sci. China Ser. A 50(2007), 1781–1794. https://doi.org/10.1007/s11425-007-0140-2.Google Scholar