## 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

### Keywords:

meromorphic function; uniqueness theorem; Nevanlinna theory

Zbl 0884.30023
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