A note on the Brück conjecture. (English) Zbl 1213.30058

Arch. Math. 95, No. 3, 257-268 (2010); erratum ibid. 99, No. 3, 255-259 (2012).
This paper is an interesting contribution to the recent literature around the R. Brück conjecture, originally formulated in [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 [J. Heittokangas, R. Korhonen, I. Laine, J. Rieppo, J. Zhang, J. Math. Anal. Appl., 355, No. 1, 352–363 (2009; Zbl 1180.30039)] and [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 \(f(z)=\frac{1}{c}(Ae^{cz}-a)+a\).


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
39A05 General theory of difference equations
Full Text: DOI


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