Total defect of a meromorphic function and all its derivatives
Let \(f\) be a meromorphic function. In addition to the usual terminology of Nevanlinna theory, put \[ \theta_ j=j \liminf_{r\to\infty}{{\bar N(r,f)} \over {T(r,f^{(j)})}}. \] Then \(0\leq\theta_ j\leq{j \over {j+1}}<1\). It is proved that \[ \delta(0,f)+\sum_{j=0}^ n \left( {1\over {1-\theta_ j}} \sum_{a\in C\setminus\{0\}} \delta(a,f^{(j)})\right) +{1 \over {1-\theta_ n}} \delta(\infty,f^{(n)})\leq 2 \] for \(n\geq 0\) and \[ \sum_{a\in C} \delta(a,f)+\sum_{j=1}^ \infty \left( {1 \over {1-\theta_ j}} \sum_{b\in C\setminus\{0\}} \delta(b,f^{(j)})\right) \leq 2. \] One of the main tools used in the proof is a result of \textit{G. Frank} and \textit{G. Weissenborn} [Bull. Lond. Math. Soc. 18, 29-33 (1988; Zbl 0586.30025)].
scientific article; zbMATH DE number 86374