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Geometric and Analytic Properties Associated With Extension Operators

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Abstract

The first aim is to prove that the Roper-Suffridge extension operator preserves \(\varepsilon \)-starlike property on general domains given by convex functions. The second is to construct the generalized Roper-Suffridge extension operator on Reinhard domains

$$\begin{aligned} \Omega _{p_{1},p_{2},\cdots ,p_{n}}=\{(z_1,\ldots ,z_{n})\in {\mathbb {C}}^{n}:\sum \limits _{j=1}^{n}|z_{j}|^{p_{j}}<1\},\quad p_{1},\cdots ,p_{n}\ge 1. \end{aligned}$$

By using a refined Schwarz-Pick lemma, we prove that the operator preserves important properties, e.g., subordination property and spirallikeness. This solves a problem of Gong and Liu. Our result improves many known results from \(p_1=2\) to \(1\le p_1<\infty \). As applications, we obtain growth and covering results associated with the extension operator on p-unit ball. Finally, by obtaining geometric and analytic properties of bounded symmetric domains, we generalize the Pfaltzgraff-Suffridge extension operator over bounded symmetric domains and prove Loewner chains and starlikeness are also preserved with a new idea. Further, we propose two conjectures for convexity property.

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Acknowledgements

The authors would like to express their deep gratitude to the referee for his/her very careful reading, valuable comments, and helpful suggestions, which made the paper more accurate and readable.

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Authors and Affiliations

  1. School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian, 362021, People’s Republic of China

    Jianfei Wang

  2. Department of Mathematics, Huzhou University, Huzhou, Zhejiang, 313000, People’s Republic of China

    Taishun Liu

  3. Department of Mathematics, Beijing Technology and Business University, Beijing, 100048, China

    Yanhui Zhang

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  1. Jianfei Wang
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  2. Taishun Liu
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Correspondence to Yanhui Zhang.

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The project was partially supported by the National Natural Science Foundation of China (Nos. 12071161, 11971165, 11971042 ) and Zhejiang Provincial Natural Science Foundation of China (No. LZ24A010004).

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Wang, J., Liu, T. & Zhang, Y. Geometric and Analytic Properties Associated With Extension Operators. J Geom Anal 34, 156 (2024). https://doi.org/10.1007/s12220-024-01600-1

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