Distribution of the \((0,\infty)\) accumulative lines of meromorphic functions (Q1344121)

Let \(f(z)\) be a meromorphic function of order \(\lambda (0 < \lambda < + \infty)\) and of lower order \(\mu\) in the plane. Two results on the distribution of the \((0, \infty)\) accumulative lines of order \(\rho\), where \(\rho\) is a positive number such that \(\mu \leq \rho \leq \lambda\), are proved in this paper. They are: (1) If the derivative \(f^{(l)} (z)\) \((0 < l < + \infty)\) has \(p(1 \leq p < \infty)\) finite nonzero deficient values \(a_i\) \((i = 1, \ldots, p)\) with deficiencies \(\delta (a_i, f^{(l)})\), then \(f(z)\) has a \((0, \infty)\) accumulative line of order \(\geq \rho\) in any angular domain with its vertex at the origin and with the magnitude greater than \[ \max \left( {\pi \over \rho},2 \pi - {4 \over \rho} \sum^p_{i = 1} \arcsin \sqrt{ {\delta (a_i, f^{(l)}) \over 2}} \right). \] (2) If \(f(z)\) has only \(p(1 \leq p < + \infty)\) \((0, \infty)\) accumulative lines of order \(\geq \rho\), for which \(\arg z = \theta_k\) \((0 \leq \theta_1 < \cdots < \theta_k < 2 \pi\), write \(\theta_{p + 1} = \theta_1 + 2 \pi)\), then \(\lambda < \pi/ \omega\), where \(\omega = \min_{1 \leq k \leq p} (\theta_{k + 1} - \theta_k)\), provided that \(f^{(l)} (z)\) \((0 \leq l < + \infty)\) has a finite nonzero deficient value.