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)+∑j=0k−1aj(z)(f○g)(j)(z)≠a(z), in Ω, is a normal family.