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) −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