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Abstract: By emplying the main ideas from G. Brosch, two results of R. Nevanlinna''s Four Value Theorem type on the unicity of meromorphic functions in the open complex plane sharing three distinct values, and a fourth value or a pair of values weakly are proved, which extend and generalize previous results of R. Nevanlinna, G.G. Gundersen, H. Ueda and G. Brosch. [Copyright &y& Elsevier]

Abstract: A uniqueness theorem for two distinct non-constant meromorphic functions that share three values of finite weights is proved, which generalizes two previous results by H.X. Yi, and X.M. Li and H.X. Yi. As applications of it, many known results by H.X. Yi and P. Li, etc. could be improved. Furthermore, with the concept of finite-weight sharing, extensions on Osgood–Yang''s conjecture and Mues'' conjecture, and a generalization of some prevenient results by M. Ozawa and H. Ueda, ect. could be obtained. [Copyright &y& Elsevier]