Hayman directions of meromorphic functions

Abstract

In this paper, we shall prove the existence of the singular directions related to Hayman's problems[1]. The results are as follows. Theorem I. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≠0, −1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$ Theorem II. Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$ Theorem III. Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive e, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$