## 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)

### MSC:

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

Zbl 0841.30027
Full Text:

### References:

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