## Meromorphic functions sharing three values or sets CM.(English)Zbl 0930.30027

In addition to the usual terminology of Nevanlinna theory, let $$\overline{N}_{k)}(r,1/(f-a))$$ and $$\overline{N}_{(k}(r,1/(f-a))$$ be the reduced counting functions of the $$a$$-points of a meromorphic function $$f$$ which have multiplicity at most $$k$$ or at least $$k$$, respectively. It is shown in Theorem 1 that if $$f$$ and $$g$$ are meromorphic functions which share 0,1,$$\infty$$ counting multiplicity and which are not Möbius transformations of each other, and if there exists $$a\not=0,1,\infty$$ such that $${T(r,f)\leq c\overline{N}_{(2}(r,{{1}\over{f-a}})+S(r,f)}$$ for some $$c>0$$, then $$f$$ and $$g$$ have the form $${f={{{e^{t\gamma}-1}\over{\lambda e^{-s\gamma}-1}}}, g={{{e^{-t\gamma}-1}\over{{{1}\over{\lambda}} e^{s\gamma}-1}}}}$$ where $$\gamma$$ is a non-constant entire function, $$\lambda$$ a non-zero constant and $$s$$ and $$t$$ are mutually prime integers, $$t>0$$. Moreover, $$\theta:=-t/s\not=1,a$$ and $${{{(1-a)^{s+t}}\over{a^t}}=\lambda^t{{(1-\theta)^{s+t}}\over{\theta^t}}}$$. Theorem 2 gives a result of this type if $${\overline{N}_{1)}(r,{{1}\over{f-a}})=S(r,f)}$$. Here $$f$$ and $$g$$ have one of nine possible forms, which are listed explicitly. Theorem 3 gives a result for functions sharing sets.

### MSC:

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

### Keywords:

shared values; meromorphic; Nevanlinna theory
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### References:

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