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Mat. Zametki, 2018, Volume 104, Issue 5, Pages 700–707 (Mi mz12168)  

Lemniscate Zone and Distortion Theorems for Multivalent Functions. II

V. N. Dubininab

a Far Eastern Federal University, Vladivostok
b Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok

Abstract: For meromorphic circumferentially mean $p$-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signed measure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.

Keywords: meromorphic function, $p$-valent function, lemniscate, Zolotarev fraction, symmetrization.

Funding Agency Grant Number
Russian Science Foundation 14-11-00022
This work was supported by the Russian Science Foundation under grant 14-11-00022.


DOI: https://doi.org/10.4213/mzm12168

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English version:
Mathematical Notes, 2018, 104:5, 683–688

Bibliographic databases:

Document Type: Article
UDC: 517.54
Received: 15.03.2018

Citation: V. N. Dubinin, “Lemniscate Zone and Distortion Theorems for Multivalent Functions. II”, Mat. Zametki, 104:5 (2018), 700–707; Math. Notes, 104:5 (2018), 683–688

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