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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
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