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Entire functions that share a polynomial with their derivatives. (English) Zbl 1096.30029

L. A. Rubel and C. C. Yang proved that if a nonconstant entire function \(f\) and its derivative \(f'\) share two distinct finite values, then \(f\equiv f'\). Jank-Mues-Volkmann proved that if a nonconstant entire function \(f\) and its derivative \(f'\) share a finite value \(a\) and \(f=a\) implies that \(f''=a\),then \(f\equiv f'\). A natural question is to consider the case that \(f\) and \(f'\) share a small function \(a(z)\) (\(T(r,a)=S(r,f)\)). In this paper, the author studied the entire functions sharing a polynomial with their derivatives. He proved that if an entire function \(f\) and its derivative \(f'\) share a nonconstant polynomial \(Q\) with degree \(q<k\) CM, and \(f(z)-Q(z)=0\) implies that \(f^{(k)}(z)-Q(z)=0\), then \(f\equiv f'\). Remark: The statement of Lemma 2 in the paper is not right. It should be stated as “Let \(f\) be an entire solution of the equation \[ a_n(z)f^{(n)}+a_{n-1}(z)f^{(n-1)}+\cdots+a_1(z)f'+a_0(z)f=0,(*) \] with polynomial coefficients \(a_0(z),\cdots,a_n(z)\) such that \(a_0(z)\not\equiv 0\) and \(a_n(z)\not\equiv 0\), then \(f\) is of finite order.” Please note that if \(a_n(z)\) is not a constant, then the solution of the equation (*) may not be an entire function.

MSC:

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