Normal families and value distribution in connection with composite functions

We prove a value distribution result which has several interesting corollaries. Let k ∈ N, let α ∈ 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 ◦ g)(k) − α 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 ∈ N, let f be a transcendental entire function with ρ(f ) < 1/k, and let a0, . . . , ak−1, a be analytic functions in a domain Ω. Then the family of analytic functions g such that (f ◦ g)(k)(z) + k−1 j=0 aj (z)(f ◦ g)(j )(z) = a(z), in Ω, is a normal family.  2005 Elsevier Inc. All rights reserved.