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Uniqueness and differential polynomials of meromorphic functions sharing a nonzero polynomial. (English) Zbl 1389.30140
Summary: Let $$k$$ be a nonnegative integer or infinity. For $$a\in\mathbb{C}\cup\{\infty\}$$ we denote by $$E_k(a;f)$$ 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 we say that $$f$$ and $$g$$ share the value $$a$$ with weight $$k$$. Using this idea of sharing values we study the uniqueness of meromorphic functions whose certain nonlinear differential polynomials share a nonzero polynomial with finite weight. The results of the paper improve and generalize the related results due to J. Xia and Y. Xu [Filomat 25, No. 1, 185–194 (2011; Zbl 1265.30161)] and the results of X.-M. Li and H.-X. Yi [Comput. Math. Appl. 62, No. 2, 539–550 (2011; Zbl 1228.30024)].
##### MSC:
 30D35 Value distribution of meromorphic functions of one complex variable, Nevanlinna theory
##### Citations:
Zbl 1265.30161; Zbl 1228.30024
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