On the zeros of \(af(f^{(k)})^n-1\) for \(n\geq 2\) (Q1880513)

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.