Abstract
In this paper we are concerned with the generalized Pfaltzgraff–Suffridge extension operator \(\Psi _{n,\alpha }\), \(\alpha \ge 0\), that provides a way of extending a locally biholomorphic mapping \(f\in H(B^n)\) to a locally biholomorphic mapping \(F\in H(B^{n+1})\). We obtain a subordination preserving result under the operator \(\Psi _{n,\alpha }\) and we consider extreme and support points associated with this operator. In the end, we present some examples of \(g\)-starlike mappings, \(g\)-spirallike mappings of type \(\alpha \) and \(g\)-almost starlike mappings of order \(\alpha \) on \(B^n\) and we consider the preservation of these notions under certain Roper–Suffridge extension operators.
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Acknowledgments
This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0899. The author is indebted to Gabriela Kohr for valuable suggestions during the preparation of this article. The author thanks the referee for suggestions and comments that improved the paper.
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Communicated by David Shoikhet.
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Chirilă, T. Extreme Points, Support Points and \(g\)-Loewner Chains Associated with Roper–Suffridge and Pfaltzgraff–Suffridge Extension Operators. Complex Anal. Oper. Theory 9, 1781–1799 (2015). https://doi.org/10.1007/s11785-015-0468-5
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DOI: https://doi.org/10.1007/s11785-015-0468-5
Keywords
- Biholomorphic mapping
- Pfaltzgraff–Suffridge extension operator
- Roper–Suffridge extension operator
- \(g\)-Loewner chain
- \(g\)-Starlike mapping
- \(g\)-Spirallike mapping of type \(\alpha \)