A note on Hayman direction for meromorphic functions of finite logarithmic order (Q5929661)

Let \(f\) be a meromorphic function in the complex plane with order \(\lambda\). If NEWLINE\[NEWLINE\limsup_{r\to+\infty}(T(r, f)/(\log r)^3) = +\infty,NEWLINE\]NEWLINE then there is a direction \(\text{arg }z= \theta\) with \(0\leq\theta< 2\pi\) such that for each \(\varepsilon\) with \(0<\varepsilon<\pi/2\), positive integer \(k\), and \(a\) and \(b\) in \(\mathbb{C}\) with \(a\neq b\), \(a,b\neq 0\), we have NEWLINE\[NEWLINE\lambda=\limsup_{r\to+\infty} \frac{\log[[n(r,\Omega(\theta,\varepsilon), f=a)+n(r,\Omega(\theta,\varepsilon),f^{(k)} = b)]\log r]}{\log\log r}NEWLINE\]NEWLINE where \(\Omega(\theta,\varepsilon) =\{z\mid|\text{arg }z-\theta|< \varepsilon\}\). The result is a quantitative version of Theorem 4.8 in \textit{L. Yang} [Value Distribution Theory (1982)] and an alternate version to a related result of \textit{H. H. Chen} [Acta Math. Sin. 30, 234-237 (1987; Zbl 0627.30028)] and is obtained through modifications of their arguments.