On the normality criteria of Montel and Bergweiler-Langley

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:448

Article

Tran Van Tan¹  Nguyen Van Thin²  Vu Van Truong³ 

[1] Hanoi Natl Univ Educ, Dept Math, 136 Xuan Thuy St, Hanoi, Vietnam

[2] Thai Nguyen Univ Educ, Dept Math, Luong Ngoc Quyen St, Thai Nguyen City, Vietnam

[3] Hoa Lu Univ, Dept Math, Ninh Binh City, Vietnam

关键词: Normal family;    Meromorphic function;    Nevanlinna theory;   

DOI  :  10.1016/j.jmaa.2016.11.008

来源: Elsevier

【 摘 要 】

A well-known result of Montel states that for a family F of meromorphic functions in a domain D subset of C, if there exist three distinct points a(1), a(2), a(3) as in C and positive integers l(1), l(2), l(3) such that 1/l(1) + 1/l(2) + 1/l(3) < 1 and all zeros of f - a(i) have multiplicity at least l(1) for all f is an element of F and i is an element of{1, 2, 3}, then F is normal in D. Inspired by this classical result, during the past 100 years, a large number of normality criteria have been established for the case where meromorphic functions (or differential polynomials generated by the members of the family) meet some distinct points with sufficiently large multiplicities. This means that these criteria strictly apply only to the case in which derivatives of functions (differential polynomials, respectively) vanish on respective zero sets. In this paper, we generalize some normality criteria of Montel, Grahl-Nevo, Gu, and Bergweiler-Langley to the case where derivatives are bounded from above on zero sets. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free