On the value distribution of ff^(k)

Abstract

Let f be a transcendental entire function. In this paper we will prove that if f is of finite order, then there exists at most one integer k{≥}2 such that ff^(k) may have non-zero and finite Picard exceptional value. We also give a class of entire functions which have no non-zero finite Picard values. If f is a transcendental meromorphic function, we obtain that for non-negative integers n, n₁, …, n_k with n{≥}1, n₁+…+n_k{≥}1, if δ(o, f)>3/(3n+3n₁+…+3n_k+1), then fⁿ(f')^n₁…(f^k)^n_k has no finite non-zero Picard values.