On angular distribution of meromorphic functions (Q1210241)

Let \(f(z)\) be a transcendental meromorphic function of order \(\lambda\), \(0<\lambda<\infty\), and lower order \(\mu\). Let \(n(r,\varepsilon,f)\) \((0<\varepsilon<\pi)\) denote the number of poles of \(f\) in \(\{x: 0\leq | z| \leq r\), \(\alpha-\varepsilon\leq\arg z\leq \alpha+\varepsilon\}\). The ray \(\arg z=\alpha\) is an accumulation line (a.c. of order \(\geq \rho\) of \(f\), if \[ \limsup_{r\to\infty}{\log[n(r,\varepsilon,f)+n(r,\varepsilon,1/f)]\over\log r}\geq\rho. \] Theorem 1. Let \(\mu\leq \rho\leq\lambda\). If \(f(z)\) has \(p\) \((1\leq p<\infty)\) deficient values \(a_ j\) \((a_ j\neq 0,\infty)\), then any sector of opening larger than \[ \max\left[{\pi\over\rho},\;2\pi- {\pi\over\rho}\sum^ p_{j=1}\arcsin\sqrt{{\delta(a_ j,f)\over 2}}\right] \] contains an a.c. of order \(\geq \rho\). Theorem 2. Let \(\mu\leq \rho\leq \lambda\). If \(\delta(a,f)>0\) \((a\neq 0,\infty)\) and if the plane is divided into \(m\) \((1\leq m<\infty)\) sectors \(S_ j\) by the a.c. of order \(\geq \rho\) of \(f(z)\), then \(\lambda\leq \pi/\omega\), where \(\omega\) is the minimum of the openings of the \(S_ j\). Theorem 3. If \(S\) is a sector of opening \({\pi\over\rho}\geq{\pi\over \lambda}\) which contains an a.c. of order \(\rho\) of \(f'(z)\), then \(S\) contains an a.e. of order \(\rho\) of \(f(z)\). These (best possible) theorems contain as special cases much previous work by Yang Lo, Zhang Guang Ho, Milloux, Gol'dberg and many others.