Angular distribution of values of \(ff'\)
type: Item
instance of: scholarly article
title: Angular distribution of values of \(ff'\)
zbMATH Open document ID: 0804.30024
author: Lo Yang
author: Chungchun Yang
published in: Science in China. Series A
publication date: 1995-01-19T00:00:00Z
review text: The main result of the authors is the following: Theorem 1. Let \(f(z)\) be an entire function of finite order \(\lambda\), \(\arg z = \theta_ j\) \((0 \leq \theta_ 1 < \theta_ 2 < \cdots < \theta_ q < 2 \pi\), \(\theta_{q + 1} = \theta_ 1 + 2 \pi)\) be a finite number of rays, \(n\) be the counting function of zeros of the function \(ff'-1\). If \(f\) satisfies \[ \varlimsup_{r \to \infty} {\log \sum^ q_{j = 1} n(r, \theta_ j + \varepsilon, \theta_{j + 1} - \varepsilon) \over \log r} \leq \rho \] for any small positive number \(\varepsilon\), then \[ \lambda \leq \max \left( {\pi \over \theta_ 2 - \theta_ 1}, \dots, {\pi \over \theta_{q + 1} - \theta_ q}, \rho \right). \] Lemma 1 (there must be changed \(O(1)\) to \(O (\log r)\) in the text of the lemma), which is cited by the authors as the result of Nevanlinna, is the hypothesis of Nevanlinna has yet to be demonstrated. I think that in place of the Lemma 1, the theorem 3.1 [\textit{A. A. Gol'dberg}, \textit{J. V. Ostrovskii}, The distribution of values of meromorphic functions. Moskva: Nauka (1970; Zbl 0217.10002) (Russian), chapter III, \S 3] is used.
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 589844
zbMATH Keywords: filling disk
zbMATH Keywords: angular characteristic function
zbMATH Keywords: angular distribution
zbMATH Keywords: derivative of entire functions
MaRDI profile type: Publication
Recommended article: Sharp bounds for the type of an entire function of order less than 1 whose zeros are located on a ray and have given averaged densities
Recommended article: The argument distribution of infinite order meromorphic functions
Recommended article: Q5464860
Recommended article: Q3743573
Recommended article: The most rapid possible growth of the maximum modulus of a canonical product of noninteger order with a prescribed majorant of the counting function of zeros
Recommended article: The Nevanlinna characteristic and maximum modulus of entire functions of finite order with random zeros
Recommended article: Q4222574
Recommended article: Q3216861
Recommended article: Two problems in the value distribution theory
Recommended article: Comparisons between T(r,f) and the total variation of \(\arg f(re^{i\theta})\) and \(\log | f(re^{i\theta})|\)
description: scientific article; zbMATH DE number 589844