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On the zeros of \(af(f^{(k)})^n-1\) for \(n\geq 2\). (English) Zbl 1056.30027

Let \(f\) be a transcendental meromorphic function and let \(a\not\equiv 0\) be a meromorphic function satisfying \(T(r,a)=S(r,f).\) One of the results obtained (Corollary 3) says that if \(n\geq2\) and \(k\geq 1,\) then \(\Psi:=a f (f^{(k)})^n-1\) has infinitely many zeros. Moreover, a lower bound for the counting function of zeros of \(\Psi\) is given. This result is in fact a special case of a more general theorem, where the derivative \(f^{(k)}\) is replaced by a linear differential polynomial.

MSC:

30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
30D30 Meromorphic functions of one complex variable (general theory)
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