Growth of logarithmic derivative of meromorphic functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1292.30015
author: Zinelaâbidine Latreuch
author: Benharrat Belaïdi
publication date: 2014-01-24T00:00:00Z
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D15
Mathematics Subject Classification ID: 34M05
zbMATH DE Number: 6250126
zbMATH Keywords: Nevanlinna theory
zbMATH Keywords: logarithmic derivative
zbMATH Keywords: meromorphic function
zbMATH Keywords: linear differential equation
zbMATH Keywords: oscillation of solutions
MaRDI profile type: Publication
OpenAlex ID: W2258208484
DOI: 10.7146/MATH.SCAND.A-15571
title: Growth of logarithmic derivative of meromorphic functions
published in: Mathematica Scandinavica
review text: We use the standard notation of value distribution theory. Let NEWLINE\[NEWLINE\rho_p(f)=\limsup _{r\to \infty} \frac{\log _{p} T(r,f)}{\log r},\quad p \in {\mathbb N}, NEWLINE\]NEWLINE where \(\log_p\) denotes the \(p\)th iteration of the logarithm, be the iterated order of a meromorphic function \(f\).NEWLINENEWLINEUsing methods of Nevanlinna theory the authors prove some new results on the growth of the logarithmic derivative.NEWLINENEWLINETheorem 1.1. Suppose that \(k\geq 2\) is an integer, and let \(f\) be a meromorphic function. Then NEWLINE\[NEWLINE \rho_1\Bigl(\frac{f'}{f}\Bigr)=\max\Bigl\{ \rho_1\Bigl(\frac{f^{(k)}}{f}\Bigr) : k\geq 2\Bigr\} .NEWLINE\]NEWLINENEWLINENEWLINENEWLINE Theorem 1.2. Let \(f\) be an entire function with a finite number of zeros. Then for any integer \(k\geq 1\) NEWLINE\[NEWLINE \rho_1\Bigl (\frac{f^{(k)}}{f}\Bigr)= \rho_2(f) .NEWLINE\]NEWLINENEWLINENEWLINEThese theorems are applied to the investigation of linear differential equations.
reviewed by: Igor Chyzhykov
description: scientific article; zbMATH DE number 6250126