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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 \((f- a)(g+ a- 1)\equiv a(1- a)\).

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