Zeros of derivatives of real meromorphic functions
type: Item
instance of: scholarly article
title: Zeros of derivatives of real meromorphic functions
zbMATH Open document ID: 1260.30016
DOI: 10.1007/BF03321825
published in: Computational Methods and Function Theory
publication date: 2012-06-25T00:00:00Z
review text: The question of characterizing those real entire functions which together with some derivative have only real zeros, or only finitely many non-real zeros, has a long history, much of which has focussed on conjectures by Pólya and Wiman. By now, many questions in this area have been answered, in particular the conjectures by Pólya and Wiman were confirmed. Much less is known for meromorphic functions. One of the results obtained here says that if \(f\) is a real meromorphic function and \(k\geq 2\), then \(f\) has the form \(f=Se^P\) with a rational function \(S\) and a polynomial \(P\) if the following four conditions are satisfied: (i) \(f\) has finitely many non-real zeros; (ii) \(f^{(k)}\) has finitely many zeros; (iii) there exists \(M\) such that the multiplicity \(m_0\) of each pole \(z_0\) satisfies \(m_0\leq M+|z_0|^M\); (iv) if \(k=2\), then \(f'/f\) has finite lower order or the non-real poles of \(f\) have bounded multiplicity. The author suggests that the result might hold without conditions (iii) and (iv). Another result deals with the distance between non-real poles of a meromorphic function \(f\) for which \(f\) and \(f''\) have only finitely many non-real zeros.
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 30D30
zbMATH DE Number: 6049595
zbMATH Keywords: real meromorphic function
zbMATH Keywords: real zero
zbMATH Keywords: Wiman conjecture
zbMATH Keywords: Laguerre-Pólya class
author: James K. Langley
MaRDI profile type: Publication
cites work: On the singularities of the inverse to a meromorphic function of finite order
cites work: Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions
cites work: Real entire functions of infinite order and a conjecture of Wiman
cites work: Nonvanishing derivatives and normal families
cites work: Zeros of Derivatives of Entire Functions
cites work: The zeros of derivatives of entire functions and the Pólya-Wiman conjecture
cites work: Bounds for the Number of Deficient Values of Certain Classes of Meromorphic Functions
cites work: Nonvanishing derivatives and the McLane class
cites work: On the Meromorphic Solutions of Non-Homogeneous Linear Differential Equations with Polynomial Coefficients
cites work: PAIRS OF LINEAR DIFFERENTIAL POLYNOMIALS
cites work: Proof of a conjecture of G. Polya concerning gap series
cites work: Estimates for the Logarithmic Derivative of a Meromorphic Function, Plus Similar Estimates
cites work: On the Characteristic of Functions Meromorphic in the Plane and of their Integrals
cites work: The Local Growth of Power Series: A Survey of the Wiman-Valiron Method
cites work: Q4692700
cites work: Derivatives of Entire Functions and A Question of Polya
cites work: Derivatives of Entire Functions and a Question of Polya. II
cites work: The Zeros of the Second Derivative of the Reciprocal of an Entire Function
cites work: Real Zeros of Derivatives of Meromorphic Functions and Solutions of Second Order Differential Equations
cites work: Q4340483
cites work: The Bergweiler-Eremenko theorem for finite lower order
cites work: On the number of nonreal zeros of real entire functions and the Fourier-Pólya conjecture
cites work: A Proof of the Polya-Wiman Conjecture
cites work: On differential polynomials, fixpoints and critical values of meromorphic functions
cites work: THE SECOND DERIVATIVE OF A MEROMORPHIC FUNCTION OF FINITE ORDER
cites work: Non-real zeros of higher derivatives of real entire functions of infinite order
cites work: Non-real zeros of derivatives of real meromorphic functions
cites work: Zeros of derivatives of meromorphic functions
cites work: Real meromorphic functions and linear differential polynomials
cites work: Non-real zeros of real differential polynomials
cites work: Non-real zeros of derivatives of real meromorphic functions of infinite order
cites work: Value distribution of differences of meromorphic functions
cites work: The Reciprocal of an Entire Function of Infinite Order ond the distribution of the Zeros of its Second Derivative
cites work: Normality criteria for families of meromorphic functions
cites work: On the zeros of the derivatives of real entire functions and Wiman's conjecture
cites work: On Borel's directions of meromorphic functions of finite order
cites work: Normal families: New perspectives
full work available at URL: https://doi.org/10.1007/bf03321825
OpenAlex ID: W4247768404
description: scientific article; zbMATH DE number 6049595