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- Title
A note on the Brück conjecture.
- Authors
Sheng Li; Zongsheng Gao
- Abstract
In this paper, we firstly consider the Brück conjecture itself and show that it holds exactly for the entire function $${f(z)=\frac{1}{c}(Ae^{cz}-a)+a}$$ , where A, a, c are nonzero constants. Then we give a necessary and sufficient condition that f( z) and f ′( z) share a finite value a CM for some special cases. Finally, we investigate two analogues of the Brück conjecture including the difference analogue of the Brück conjecture raised by Liu and Yang (Arch. Math. 92, 270–278 (2009)) and the shifted analogue of the Brück conjecture raised by Heittokangas et al. (J. Math. Anal. Appl. 355, 352–363 (2009)). And we give some necessary conditions when f( z) shares a finite value a CM with its difference operators or shifts.
- Subjects
LOGICAL prediction; LINEAR operators; COMPLEX variables; MATHEMATICS; DIFFERENCE algebra; DIFFERENCE equations; DIFFERENTIAL algebra
- Publication
Archiv der Mathematik, 2010, Vol 95, Issue 3, p257
- ISSN
0003-889X
- Publication type
Article
- DOI
10.1007/s00013-010-0165-6