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Sharing three values with small weights. (English) Zbl 1146.30015
Let $$f$$ and $$g$$ be two nonconstant meromorphic functions defined in the open complex plane $$\mathbb{C}$$. For $$b\in\mathbb{C}\cup\{\infty\}$$ we say that $$f$$ and $$g$$ share the value $$b$$ CM (counting multiplicities) if $$f$$ and $$g$$ have the same $$b$$-points with the same multiplicities. If we do not take multiplicities into account, we say that $$f$$ and $$g$$ share the value $$b$$ IM (ignoring multiplicities).
Let $$p\in\mathbb{N}$$ and $$b\in\mathbb{C}\cup\{\infty\}$$. By $$N(r,b; f|\leq p)$$ we denote the counting function of those $$b$$-points of $$f$$ whose multiplicities are not greather then $$p$$ (counting with proper multiplicities). By $$\overline N(r,b;f|\leq p)$$ we denote the corresponding reduced counting function.
Analogously we define numbers $$N(r,b;f|\geq p)$$ and $$\overline N(r,b;|\geq p)$$.
In this paper the following result is proved:
Theorem: Let $$f$$ and $$g$$ be two distinct meromorphic functions sharing $$(0,1)$$, $$(1, m)$$ and $$(\infty, k)$$, where $$(m- 1)(mk- 1)> (1+ m)^2$$. If $$a(\neq 1,0)$$ is a finite complex number such that $$N(r,a;f|\leq 2)\neq T(r,f)+ S(r,f)$$ and $$N(r,\infty;|\leq 1)\neq T(r,f)+ S(r,f)$$ then $$a$$ and $$a- 1$$ are Picard exceptional values of $$f$$ and $$g$$ respectively and also $$\infty$$ is so and $$(f- a)(g+ a- 1)\equiv a(1- a)$$.

##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Keywords:
meromorphic function
Full Text:
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