On the zeros of a differential polynomial and normal families (Q1353798)

Let \(f\) be a meromorphic function in the complex plane, \(k\) a nonnegative integer, \(n=k+3\). Let \(a_j\) \(j=1,2,3,\dots,k+1)\) be complex numbers, and assume \(\Psi=f^{(k)}+a_1f^{(k-1)}+ \cdots+ a_kf+a_{k+1}\neq 0\). Let \(Q=f^n+\Psi\) and assume, using the usual Nevanlinna theory notation, that \(N(r,1/Q)= S(r,f)\). Then \(f\) is shown to satisfy an explicit Riccati differential equation. Further, if \(n\) is odd and \(a_1=0\) or if \(n\) is even and \(a_1=a_2=0\), \(f\) is constant. These results generalize work of \textit{W. K. Hayman} [Ann. Math., II. Ser., 70, 9-42 (1959; Zbl 0088.28505)], \textit{E. Mues} [Math. Z. 164, No. 3, 239-259 (1979; Zbl 0402.30034); and \textit{N. Steinmetz} [Math. Z. 176, No. 2, 255-264 (1981; Zbl 0466.30026)]. The proof relies heavily on Steinmetz's work and a modified version of a lemma of \textit{J. Clunie} [J. Lond. Math. Soc. 37, 17-27 (1962; Zbl 0104.29504)]. The authors also give a normal families theorem of the nature suggested by the Bloch principle applied to the results above.