Extension Operators for Biholomorphic Mappings

Wang, Jianfei ; Zhang, Danli

Canadian mathematical bulletin, Tome 62 (2019) no. 3, pp. 671-679

Résumé

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.

Détail

DOI : 10.4153/CMB-2018-021-0

Mots-clés : biholomorphic mapping, ε-starlike mapping, spiral-like mapping

Bibliographie

[1] 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

[2] Elin, M., Extension operators via semigroups . J. Math. Anal. Appl. 377(2011), 239–250. . Google Scholar | DOI

[3] Elin, M. and Levenshtein, M., Covering results and perturbed Roper-Suffridge operators . Complex Anal. Oper. Theory. 8(2014), 25–36. . Google Scholar | DOI

[4] Feng, S. and Liu, T., The generalized Roper-Suffridge extension operator . Acta Math. Sci. Ser. B 28(2008), 63–80. . Google Scholar | DOI

[5] Feng, S. and Yu, L., Modified Roper-Suffridge operator for some holomorphic mappings . Front. Math. China 6(2011), 411–426. . Google Scholar | DOI

[6] Gong, S. and Liu, T., On Roper-Suffridge extension operator . J. Anal. Math. 88(2002), 397–404. . Google Scholar | DOI

[7] Gong, S. and Liu, T., The generalized Roper-Suffridge extension operator . J. Math. Anal. Appl. 284(2003), 425–434. . Google Scholar | DOI

[8] Graham, I., Hamada, H., Kohr, G., and Suffridge, T. J., Extension operators for locally univalent mappings . Michigan. Math. J. 50(2002), 37–55. . Google Scholar | DOI

[9] 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

[10] Graham, I. and Kohr, G., Univalent mappings associated with the Roper-Suffridge extension operator . J. Anal. Math. 81(2000), 331–342. . Google Scholar | DOI

[11] Liu, M. and Zhu, Y., On some sufficient conditions for starlikeness of order 𝛼 in ℂ n . Taiwanese J. Math. 10(2006), 1169–1182. . Google Scholar | DOI

[12] Liu, X., The generalized Roper-Suffridge extension operator for some biholomorphic mappings . J. Math. Anal. Appl. 324(2006), 604–614. . Google Scholar | DOI

[13] Liu, T. and Xu, Q., Loewner chains associated with the generalized Roper-Suffridge extension operator . J. Math. Anal. Appl. 322(2006), 107–120. . Google Scholar | DOI

[14] Roper, K. and Suffridge, T. J., Convex mappings on the unit ball of ℂ n . J. Anal. Math. 65(1995), 333–347. . Google Scholar | DOI

[15] Wang, J., Modified Roper-Suffridge operator for some subclasses of starlike mappings on Reinhardt domains . Acta Math. Sci. Ser. B 33(2013), 1627–1638. . Google Scholar | DOI

[16] Wang, J. and Liu, T., The Roper-Suffridge extension operator and its applications to convex mappings in ℂ2 . Trans. Amer. Math. Soc. . Google Scholar | DOI

[17] Xu, Q., Liu, T., and Liu, X., On a subclass of close-to-convex mappings . Complex Anal. Oper. Theory. 9(2015), 275–286. . Google Scholar | DOI

[18] Zhu, Y. and Liu, M., The generalized Roper-Suffridge extension operator on bounded complete Reinhardt domains . Sci. China Ser. A 50(2007), 1781–1794. . Google Scholar | DOI