An extension of Picard's theorem for meromorphic functions of small hyper-order

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:357

Article

Korhonen, Risto

关键词: Picard's theorem;    Second main theorem;    Hyper-order;    Forward invariant;    Value distribution;   

DOI  :  10.1016/j.jmaa.2009.04.011

来源: Elsevier

【 摘 要 】

A version of the second main theorem of Nevanlinna theory is proved, where the ramification term is replaced by a term depending on a certain composition operator of a meromorphic function of small hyper-order. As a corollary of this result it is shown that if n is an element of N and three distinct values of a meromorphic function f of hyper-order less than 1/n(2) have forward invariant pre-images with respect to a fixed branch of the algebraic function tau(z) = z + alpha(n-1)z(1-1/n) + ... + alpha(1)z(1/n) + alpha(0) with constant coefficients, then f circle tau equivalent to f. This is a generalization of Picard's theorem for meromorphic functions of small hyper-order, since the (empty) pre-images of the usual Picard exceptional values are special cases of forward invariant pre-images. (C) 2009 Elsevier Inc. All rights reserved.

【 授权许可】

Free