Meromorphic functions that share one or two values. II. (English) Zbl 0939.30020

For a meromorphic function \(h\), let \(\overline N_{(2}(r,h)\) denote the counting function of multiple poles of \(h\), each counted only once. Define \(N_2(r,h) = \overline N(r,h) + \overline N_{(2}(r,h)\) and \(N^*(r,h) = 2N_2(r,h) + 3\overline N(r,h)\). The author proved in [Complex Variables, Theory Appl. 28, 1-11 (1995; Zbl 0841.30027)], that either \(f\equiv g\) or \(fg\equiv 1\) whenever \(f, g\) are two nonconstant meromorphic functions sharing the value \(1\) CM and in a set \(I\subset [0, +\infty)\) of infinite linear measure, \[ \limsup_{r\to \infty, r\in I}\frac{N_2(r,1/f) + N_2(r,f) + N_2(r, 1/g) + N_2(r,g)}{\max(T(r,f), T(r,g))}<1. \] In this paper, the same conclusion will be proved for \(f, g\) sharing the value \(1\) IM, provided \[ \limsup_{r\to\infty, r\in I}\frac{N^*(r,1/f) + N^*(r,f) + N^*(r,1/g) + N^*(r,g)}{T(r,f) + T(r,g)}<1. \] If \(f,g\) share the values \(1\) and \(\infty\) IM and \[ \limsup_{r\to \infty, r\in I} \frac{N^*(r,1/f) + N^*(r,1/g) + 12\overline N(r,f)}{T(r,f) + T(r,g)}<1, \] then again \(f\equiv g\) or \(fg\equiv 1\). The proofs need a careful analysis of \(1\)-points of \(f\) and \(g\) with respect to different multiplicities combined with elementary Nevanlinna theory.
Reviewer: I.Laine (Joensuu)


30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory


Zbl 0841.30027
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