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