Unique factorizability of the pth power of entire functions (Q805794)

The author proves two interesting theorems on unique factorization in the composition sense: Theorem 1. Let f(z) be a prime transcendental function with infinitely many zeros \(\{z_ n\}\) all of which satisfy \(| z_{n+1}| \geq k| z_ n|\) \((n=1,2,...)\) for some \(k>1\). Then \(f^ p(z)\) is uniquely factorizable for every prime p. Theorem 2. Let f(z) be a prime transcendental entire function with finitely many zeros. Then \(f^ p(z)\) is uniquely factorizable, unless \[ f^ p(z)=g((z+c)^ p), \] where g(w) is an entire function and c is a constant. The reviewer initially raised questions on the uniqueness of factorization even of products of prime functions in his book: Factorization of meromorphic functions and subsequent publications.