A hypothesis of Littlewood and the distribution of values of entire functions
type: Item
instance of: scholarly article
title: A hypothesis of Littlewood and the distribution of values of entire functions
zbMATH Open document ID: 0601.30039
DOI: 10.1007/BF01077319
publication date: 1986-01-01T00:00:00Z
review text: Let \(\rho (f)=| f'| /(1+| f|^ 2)\) be the spherical derivative of a polynomial f. Denote \[ \phi (n)=\sup \{\iint_{x^ 2+y^ 2\leq 1}\rho (f) dx dy: \deg f=n\}. \] J. Littlewood proved that \(\phi\) (n)\(\leq \pi \sqrt{n}\) and conjectured that \(\phi (n)\leq An^{\alpha}\) with absolute constants \(A>0\), \(\alpha <\). The authors prove that \(\phi (n)=o(n)\), \(n\to \infty\). This result has an interesting counterpart in the theory of entire functions: let f be an entire function of finite positive order. Then there exists an infinitesimal part of the plane which contains almost all solutions of \(f(z)=a\) for all \(a\in {\mathbb{C}}\).
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 31A05
zbMATH DE Number: 3969015
zbMATH Keywords: subharmonic function
zbMATH Keywords: Riesz measure
zbMATH Keywords: spherical derivative
author: Alexandre Eremenko
author: M. I. Sodin
MaRDI profile type: Publication
cites work: On some Conjectural Inequalities, with Applications to the Theory of Integral Functions
cites work: On a conjecture of Littlewood
cites work: Null Sets for Measures Orthogonal to R(X)
full work available at URL: https://doi.org/10.1007/bf01077319
OpenAlex ID: W2046762746
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 3969015