A note on the Brück conjecture
type: Item
instance of: scholarly article
title: A note on the Brück conjecture
zbMATH Open document ID: 1213.30058
author: Sheng Li
author: Zongsheng Gao
published in: Archiv der Mathematik
publication date: 2010-10-08T00:00:00Z
review text: This paper is an interesting contribution to the recent literature around the R. Brück conjecture, originally formulated in [\textit{R. Brück,} Results Math., 30, No.1--2, 21--24 (1996; Zbl 0861.30032)]. The key results here are related to the derivative version, the shift version and the difference operator version of the conjecture. As to the last two of these versions, see [\textit{J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang}, J. Math. Anal. Appl., 355, No. 1, 352--363 (2009; Zbl 1180.30039)] and [\textit{K. Liu, L.-Z. Yang}, Arch. Math., 92, No. 3, 270--278 (2009; Zbl 1173.30018)], respectively. The main result for the difference operator version reads as follows: Let \(f(z)\) be a non-periodic transcendental entire function of finite order \(\rho\), and suppose that \(f(z)\) and the iterated difference operator \(\Delta_{\eta}^{n}f(z)\) share a nonzero finite complex value \(a\) CM. Then \(1\leq\rho\leq \lambda (f-a)+1\), where \(\lambda (f-a)\) stands for the exponent of convergence for the sequence of \(a\)-points of \(f\). The corresponding result for the shift version is more complicated. The reviewer remarks that the abstract of the paper is somewhat misleading. Indeed, what is actually observed in the paper, is that the Brück conjecture (in its original derivative version) is equivalent to the following conjecture: Let \(f(z)\) be a non-constant entire function such that the hyper-order \(\rho_{2}(f)<\infty\) is not a natural number, and suppose that \(f(z),f'(z)\) share a finite value \(a\) CM. Then there exist nonzero constants \(A,c\) such that \(f(z)=\frac{1}{c}(Ae^{cz}-a)+a\).
Mathematics Subject Classification ID: 30D35
Mathematics Subject Classification ID: 39A05
zbMATH DE Number: 5797320
zbMATH Keywords: Brück conjecture
zbMATH Keywords: derivative
zbMATH Keywords: shift
zbMATH Keywords: difference
zbMATH Keywords: shared value
Wikidata QID: Q123291322
reviewed by: Ilpo Laine
MaRDI profile type: Publication
full work available at URL: s00013-010-0165-6
OpenAlex ID: W2017673442
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DOI: 10.1007/S00013-010-0165-6
description: scientific article; zbMATH DE number 5797320