Hayman directions of meromorphic functions

In this paper, we shall prove the existence of the singular directions related to Hayman's problems^[1]. The results are as follows.

1. 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 ε, every integer p(≠0, ?1) and every finite complex number b(≠0), we have $$mathop {lim }limits_{r to infty } left{ {n(r,theta _0 ,varepsilon ,f' cdot { f} ^p = b)} right} = + infty $$
2. 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 ε, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$mathop {lim }limits_{r to infty } left{ {n(r,theta _0 ,varepsilon ,f' - a{ f} ^p = b)} right} = + infty $$
3. 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 ε, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$mathop {lim }limits_{r to infty } left{ {n(r,theta _0 ,varepsilon ,f' - a{ f} ^p = b)} right} = + infty $$

The singular directions in Theorems I–III are called Hayman directions.