Some quantitative estimations on the zeros of differential polynomials (Q1313684)

This article presents some estimates for the counting functions of zeros of differential polynomials. These estimates are related with earlier well-known results due to Hayman, Clunie and Mues. We quote a couple of typical results: (1) Let \(f\) be a transcendental meromorphic function and let \(Q(f)\) be a nonzero differential polynomial of \(f\) of degree \(n-1\). Suppose that \(\Psi=f^ nf'+Q(f)\). Then \(T(r,f) \leq 8 \overline N(r,{1 \over \Psi})+(5\alpha+9) \overline N(r,f)+S(r,f)\), where \(\alpha=\Gamma_ Q-(n-1)\), \(\Gamma_ Q\) the weight of \(Q\). (2) Let \(f\) be a transcendental meromorphic function, let \(P(f)=a_ nf^ n+ \cdots+a_ 0\) be a polynomial in \(f\) with small coefficients (in the sense of Nevanlinna theory) such that \(a_{n-1}/a_ n\) is constant, and let \(Q(f)\) be a differential polynomial of \(f\) of order \(\leq n-1\). Suppose that \(\Psi=P(f)f'+Q(f)\). Then either \(\Psi =a_ n(f+{a_{n-1} \over na_ n})^ nf'\) or \(T(r,f) \leq 8 \overline N(r,{1 \over \Psi})+(5\beta+9) \overline N(r,f)+ S(r,f)\), where \(\beta=\max (1,\Gamma_ Q-(n-1))\).