Normal families and value distribution in connection with composite functions

[摘要] We prove a value distribution result which has several interesting corollaries. Let k epsilon N, let alpha epsilon C and let f be a transcendental entire function with order less than 1/2. Then for every nonconstant entire function g, we have that (f circle g) ((k)) - alpha has infinitely many zeros. This result also holds when k = 1, for every transcendental entire function g. We also prove the following result for normal families. Let k epsilon N, let f be a transcendental entire function with rho (f) < 1/k, and let a(0),..., a(k-1), a be analytic functions in a domain Omega 2. Then the family of analytic functions g such that (f circle g)((k))(z) + Sigma(j=0) a(j) (z) (f circle g) ((j)) (z) not equal a (z), in Omega, is a normal family. (c) 2005 Elsevier Inc. All rights reserved.

[发布日期] 2005-12-01

[关键词] normal families;value distribution;composite functions;Nevanlinna theory