Deficient rational functions and Ahlfors's theory of covering surfaces
Let \(f(z)\) be a transcendental meromorphic function in the plane, and let \(a_k\), \(k=1,\dots, q\) be distinct complex numbers or \(\infty\). The author considers a generalization of \textit{R. Nevanlinna's} second fundamental theorem \[ (q-2)T(r,f)\leq\sum_{k=1}^q\overline{N}\bigg(r,\frac{1}{f-a_k}\bigg)+S(r,f).\leqno{(1)} \] for distinct small functions. \textit{N. Steinmetz } proved [J. Reine Angew. Math. 368, 134-141 (1986; Zbl 0598.30001) ] \[ (q-2-\varepsilon)T(r,f)\leq\sum_{k=1}^q{N}\bigg(r,\frac{1}{f-a_k}\bigg)+S(r,f),\leqno{(2)} \] for distinct small functions, see also \textit{C. F Osgood} [J. Number theory 21, 347-389 (1985; Zbl 0575.10032)], \textit{G. Frank} and \textit{G. Weissenborn} [Bull. Lond. Math. Soc. 18, 29-33 (1986; Zbl 0586.30025 ]. It has been an open question whether (2) holds if \(N\) is replaced by \(\overline{N}\). The author proves that for rational functions \(R_k\), \(k=1,\dots, q\) with distinct values at \(\infty\), \(a_k\) in (1) can be replaced by \(R_k\). The key idea for the proof is the combination of \textit{Alhfors'} second fundamental theorem and \textit{Rouché's} theorem.
scientific article; zbMATH DE number 1928898
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