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Meromorphic functions sharing zeros and poles and also some of their derivatives sharing zeros. (English) Zbl 0637.30029
If two meromorphic functions f and g share 0 and \(\infty CM\), then \(g=f \exp \beta\) for a certain entire function \(\beta\). In this paper we show: If, for \(n=1,...,6\), the derivatives \(f^{(n)}\) and \(g^{(n)}\) share 0 CM, then \(\beta\) is constant, unless f and g have space representations. If f and g are of finite order, if f’ and g’ share 0 CM, and if, for one \(n>1\), \(f^{(n)}\) and \(g^{(n)}\) share 0 CM, then we obtain the same result. Thus problem 2.65 raised by A. Hinkkanen is solved [K. F. Barth, D. A. Brannan and W. K. Hayman, Bull. Lond. Math. Soc. 16, 490-517 (1984; Zbl 0593.30001)].
Reviewer: L.Köhler

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