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Weighted sharing of three values and uniqueness of meromorphic functions. (English) Zbl 0996.30020
Let $$f$$ and $$g$$ be distinct nonconstant meromorphic functions in the complex plane, $$k$$ a nonnegative integer or infinity, and $$a$$ a member of $$\mathbb{C}\cup \{\infty\}$$. Let $$E_k(a;f)$$ be the set of all $$a$$-points of $$f$$ where an $$a$$-point of multiplicity $$m$$ is counted $$m$$ times if $$m\leq k$$ and $$k+1$$ times if $$m>k$$. If $$E_k(a;f)= E_k(a;g)$$, then $$f$$ and $$g$$ are said to share the value $$a$$ with weight $$k$$. Using weighted sharing the author proves results on the uniqueness of meromorphic functions sharing three values in the complex plane which improve theorems of H. Ueda [Kodai Math. J. 3, 457-471 (1980; Zbl 0468.30023)], H. X. Yi [Chin. Ann. Math., Ser. A 9, 434-439 (1988; Zbl 0699.30024)], and S. Z. Ye [Kodai Math. J. 15, 236-243 (1992; Zbl 0767.30026)].
Reviewer: L.R.Sons (DeKalb)

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