On the theorem of Tumura-Clunie (Q1124005)

Let f be meromorphic in the plane. A meromorphic function g is called ``small'' (with respect to f) if its Nevanlinna characteristic satisfies \(T(r,g)=S(r,f)\). A differential polynomial Q[f] is a finite sum of monomials \(a(z)(f)^{j_ 0}(f')^{j_ 1}...(f^ n)^{j_ n},\) a ``small''. The numbers \(\gamma_ Q=\max (j_ 0+j_ 1+...+j_ n)\) and \(\Gamma_ Q=\max (j_ 0+2j_ 1+...+(n+1)j_ n)\) are called the degree and the weight of Q, respectively. Generalizing a theorem of Tumura, which was first proved by \textit{J. Clunie} [J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.295)] and, in a refined form, by \textit{W. K. Hayman} [Meromorphic functions (1964; Zbl 0149.030)] the author proves the following Theorem. Let Q be a differential polynomial of degree \(\leq n-2\) and set \[ F=f^ n+a_{n-1}f^{n-1}+Q[f]. \] Then, either \(F=(f+a_{n- 1}/n)^ n\) or \[ 2T(r,f)\leq \bar N(r,1/F)+\alpha \bar N(r,f)+\bar N(1,(f+a_{n-1}/n)^{-1})+S(r,f), \] where \(\alpha =\max (1,\Gamma_ Q+3-n)\).