On the number of real critical points of logarithmic derivatives and the Hawaii conjecture
type: Item
instance of: scholarly article
title: On the number of real critical points of logarithmic derivatives and the Hawaii conjecture
zbMATH Open document ID: 1247.30009
author: Mikhail Tyaglov
published in: Journal d'Analyse Mathématique
publication date: 2012-05-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/0902.0413
review text: A real entire function \(f\) belongs to the class \(V_{2n}\) if \(f(z) = \exp(-az^{2n+2})g(z)\), where \(a \geq 0\) and \(g\) is a real entire function of genus \(\leq 2n + 1\) with only real zeros. The class \(U_{2n}\) can be defined recursively as \(U_0 = V_0\) and \(U_{2n} = V_{2n} - V_{2n-2}\) when \(n\geq 1\). A function \(\varphi\) is in the class \(U_{2n}^*\) whenever there exist \(f\in U_{2n}\) and a real polynomial \(p\) with no real zeros such that \(\varphi=p f\). The class \(U_{0}\) is called the Laguerre-Pólya class and is usually denoted by \(\mathcal{L}\text{-}\mathcal{P}\). The class \(U_{0}^*\) is frequently denoted by \(\mathcal{L}\text{-}\mathcal{P}^*\). For a given entire function \(\varphi \in U_{2n}^*\), the author establishes a connection between the number of real zeros of the functions \(Q[\varphi]:=(\varphi^{\prime}/\varphi)^{\prime}\) and \(Q_1[\varphi]:=(\varphi^{\prime\prime}/\varphi^{\prime})^{\prime}\). This investigation enables him to prove the Hawaii conjecture, first formulated by \textit{T. Craven}, the reviewer and \textit{W. Smith} [``The zeros of derivatives of entire functions and the Pólya-Wiman conjecture'', Ann. Math. (2) 125, 405--431 (1987; Zbl 0625.30036)]. The Hawaii conjecture states that the number of non-real zeros of a real polynomial \(p\) is not less than the number of real critical points of \(p^{\prime}/p\). In addition, the author shows that if the real entire function \(\varphi \in \mathcal{L}\text{-}\mathcal{P}^*\) has \(2m\) non-real zeros (counting multiplicities) and \(\varphi^{\prime}\) has \(2m_1\) non-real zeros, then \(Z_R(Q)\), the number of real zeros of \(Q[\varphi]\), satisfies \(2m-2m_1\leq Z_R(Q)\leq 2m\). This proves the generalized Hawaii conjecture. The author also derives several interesting results pertaining to the number of real critical points of the logarithmic derivatives of functions in \(U_{2n}^*\) (cf. Section 5).
reviewed by: George Csordas
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 26C10
zbMATH DE Number: 6031949
zbMATH Keywords: Hawaii conjecture
zbMATH Keywords: critical points
zbMATH Keywords: logarithmic derivative
zbMATH Keywords: Laguerre-Pólya class
MaRDI profile type: Publication
OpenAlex ID: W2052472665
Wikidata QID: Q123189791
arXiv ID: 0902.0413
cites work: Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions
cites work: Q5833361
cites work: Q4658944
cites work: The zeros of derivatives of entire functions and the Pólya-Wiman conjecture
cites work: Zeros of Derivatives of Entire Functions
cites work: Linear operators and the distribution of zeros of entire funtions
cites work: Real Wronskian zero of polynomials with nonreal zeros
cites work: Zeros of the Wronskian of a polynomial
cites work: Non-real Zeros of Derivatives of Real Entire Functions and the Pólya-Wiman Conjectures
cites work: Level sets, a Gauss-Fourier conjecture, and a counter-example to a conjecture of Borcea and Shapiro
cites work: Derivatives of Entire Functions and A Question of Polya
cites work: Q5531193
DOI: 10.1007/S11854-011-0011-1
description: scientific article; zbMATH DE number 6031949