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Mat. Zametki, 2010, Volume 87, Issue 6, Pages 848–854 (Mi mz7700)  

On Holomorphic Motions of $n$-Symmetric Functions

I. R. Kayumov

Kazan State University

Abstract: We generalize a problem examined by Duren on the univalence of a family of $n$-symmetric functions generated by integrals of functions of the form $\exp(\lambda \zeta^n)$. Our approach is based on the use of the inverse Faber transform, of the Martio–Sarvas univalence criterion, and of the $\lambda$-lemma of Mañé, Sad, and Sullivan. We also put forward a conjecture on the univalence of a family of $n$-symmetric functions, which is a weakened form of the Danikas–Ruscheweyh conjecture on the univalence of an integral transform of holomorphic functions.

Keywords: $n$-symmetric function, inverse Faber transform, domain with quasiconformal boundary, Danikas–Ruscheweyh conjecture, holomorphic function

DOI: https://doi.org/10.4213/mzm7700

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English version:
Mathematical Notes, 2010, 87:6, 828–833

Bibliographic databases:

UDC: 517.54
Received: 19.02.2010
Revised: 22.04.2010

Citation: I. R. Kayumov, “On Holomorphic Motions of $n$-Symmetric Functions”, Mat. Zametki, 87:6 (2010), 848–854; Math. Notes, 87:6 (2010), 828–833

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