Two-point distortion theorems and the Schwarzian derivatives of meromorphic functions

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:467

Article

Dubinin, V. N.^1,2 

[1] Far Eastern Fed Univ, Vladivostok, Russia

[2] FEBRAS, Inst Appl Math, 7 Radio St, Vladivostok, Russia

关键词: Two-point distortion;    Schwarzian derivative;    Meromorphic function;    Condenser capacity;   

DOI  :  10.1016/j.jmaa.2018.07.007

来源: Elsevier

【 摘 要 】

For a meromorphic function f in the unit disk U = {z : vertical bar z vertical bar < 1} and arbitrary points z(1), z(2) in U distinct from the poles of f, a sharp upper bound on the product vertical bar f'(z(1))f'(z(2))vertical bar is established. Further, we prove a sharp distortion theorem involving the derivatives f'(z(1)), f'(z(2)) and the Schwarzian derivatives S-f(z(1)), S-f(z(2)) for z(1), z(2) is an element of U. Both estimates hold true under some geometric restrictions on the image f(U). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free