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Entire functions sharing a polynomial with their derivatives and normal families. (English) Zbl 1135.30312

Summary: Let \(f\) be a nonconstant entire function and \(Q\) be a nonconstant polynomial. Let \(M\) be a positive number and \(k\geq 2\) be an integer. We prove that if \(f\) and \(f'\) share \(Q\) CM and \(|f^{(k)}(z)|\leq M\cdot (1+|Q(z)|)\) whenever \(f(z)=Q(z)\), then \(\frac{f'-Q}{f-Q}\) is constant. Furthermore, we prove the corresponding normality criterion.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D45 Normal functions of one complex variable, normal families
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