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Meromorphic functions sharing a nonzero value with their derivatives. (English) Zbl 1320.30060

Summary: Let \(f\) be a transcendental meromorphic function of finite order in the plane such that \(f^{(m)}\) has finitely many zeros for some positive integer \(m\geq 2\). Suppose that \(f^{(k)}\) and \(f\) share \(a\) CM, where \(k\geq1\) is a positive integer, \(a\not=0\) is a finite complex value. Then \(f\) is an entire function such that \(f^{(k)}-a=c(f-a)\), where \(c\not=0\) is a nonzero constant. The results in this paper are concerning a conjecture of R. Brück [Result. Math. 30, No.1–2, 21–24 (1996; Zbl 0861.30032)]. An example is provided to show that the results in this paper, in a sense, are the best possible.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory

Citations:

Zbl 0861.30032
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