Meromorphic functions that share a nonzero polynomial with finite weight
Using the notion of weighted sharing the author proves two results on uniqueness of meromorphic functions when two nonlinear differential polynomials share a polynomial. The theorems improve some existing results. One typical result of the paper is the following: Let \(f\) and \(g\) be two transcendental meromorphic functions, let \(n \geq 11\) be a positive integer and \(P \not \equiv 0\) be a polynomial with degree \(\gamma_{P} \leq 11\). If \(f^{n}f^{\prime} - P\) and \(g^{n}g^{\prime} - P\) share \((0, 2)\), then either \(f = tg\) for a complex number \(t\) satisfying \(t^{n + 1} = 1\), or \(f = c_{1}e^{cQ}\) and \(g = c_{2}e^{-cQ}\), where \(c_{1}\), \(c_{2}\), \(c\) are three nonzero complex numbers satisfying \((c_{1}c_{2})^{n + 1}c^{2} = -1\) and \(Q\) is a polynomial satisfying \(Q = \int_{0}^{z}P(\eta)d\eta\).
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