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On a theorem of Ozawa. (English) Zbl 0919.30021
Let $$f$$ and $$g$$ be nonconstant meromorphic functions in the complex plane. Let $$E(a,f)$$ be the set of points counting multiplicity at which $$f$$ assumes $$a\in\mathbb{C}\cup\{\infty\}$$. Assuming $$E(1,f)=E(1,g)$$, $$E(\infty,f)= E(\infty,g)$$, and $$u=\lambda\Theta (\infty,f)+ (z-\lambda) \Theta (\infty,g)+ \delta_2(0,f)+\delta_2 (0,g)>3$$ where $$0\leq\lambda\leq 2$$, the author shows $$f \equiv g$$ or $$fg\equiv 1$$. Here $\delta_2(a,f)=1-\lim_{r\to\infty} \sup\left( \left( \left.\overline N\bigl(r,1/(f-a) \bigr)+ \overline N_{(2}\left( r,{1\over f-a}\right)\right)\right /T(r,f) \right)$ where $$N_{(2}\bigl(r,1/(f-a)\bigr)$$ counts only multiple zeros of $$f(z)-a$$. The results improve theorems of H.-X. Yi [Kodai Math. J. 13, 363-372 (1990; Zbl 0712.30029)] and M. Ozawa [J. Anal. Math. 30, 411-420 (1976; Zbl 0337.30020)].
Reviewer: L.R.Sons (DeKalb)

##### MSC:
 30D30 Meromorphic functions of one complex variable, general theory