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Linear differential polynomials sharing three values with weights. (English) Zbl 1031.30016

Let \(f\) and \(g\) be two nonconstant meromorphic functions in \(\mathbb{C}\) and let \(L(y)=\sum_{i=1}^p\alpha_i\frac{d^iy}{dz^i}\) be a linear differential operator with constant coefficients. The author proves the following result: If \(L(f)\) and \(L(g)\) are nonconstant such that \(L(f)\) and \(L(g)\) share values \(0\) and \(1\) truncated to the multiplicity \(2\), and such that the Nevanlinna defects \(\delta(a,f)\) and the defects \(\delta_p(a,f)\) truncated to the multiplicity \(p\) satisfy \(\sum_{a\not=\infty}\delta_p(a,f)>0\) and \[ \frac{\sum_{a\not=\infty}\delta(a,f)}{1+p(1-\delta_1(\infty,f))}> \frac{1}{2}+ \frac{\lambda(1-\delta_1(\infty,f))}{\sum_{a\not=\infty}\delta_p(a,f)}, \] and if \(f\) and \(g\) share \(\infty\) IM (or CM) with \(\lambda=2\) (or \(\lambda=1\)), then either \(L(f)L(g)=1\) or \(f-g=q\), where \(q\) is a solution of the differential equation \(L(y)=0\). This improves a result of the author [Yokohama Math. J. 44, 147-156 (1997; Zbl 0884.30023)].
Reviewer: Pei-Chu Hu (Jinan)

MSC:

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

Citations:

Zbl 0884.30023
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