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On a result of Ozawa and uniqueness of meromorphic function. (English) Zbl 1160.30018

In Nevanlinna theory for a meromorphic function \(f\) in \(\mathbb{C}\), \(T(r,f)\), \(N(r,f)\), \(\overline N(r,f)\) etc. are defined. For \(a\in \mathbb{C}\cup\{\infty\}\), let \(E_k(a, f)\) denote the set of all zeros of \(f- a\) \((a\neq\infty)\) or \(1/f\) \((a=\infty)\), where a zero with multiplicity is counted \(m\) times if \(m\leq k\) and \(k+1\) times if \(m>k\). Let \(N_{(2}(r,{1\over f-a})\) be the counting function of the zeros of \(f- a\) \((a\neq\infty)\) with multiplicity \(\geq 2\), and let \(\overline N_{(2}(r,{1\over f-a})\) be the corresponding reduced counting function. Let \[ N_2\Biggl(r,{1\over f-a}\Biggr)= \overline N\Biggl(r,{1\over f-a}\Biggr)+\overline N_{(2}\Biggl(r,{1\over f-a}\Biggr) \] and \[ \delta_2(a,f)= f-\varlimsup_{r\to\infty} {N_2(r,{1\over f-a})\over T(r,f)}. \] The author proves the following theorem and two corollaries which improve some previous results:
Theorem. If \(f\) and \(g\) are two meromorphic functions in \(\mathbb{C}\) such that \(E_p(a, f)= E_p(a,g)\) for \((a,p)= (0,m)\) \((m\geq 2)\), \((\infty, 0)\) and \((1,1)\) and \[ 2\delta_2(0, f){4m\over m-1} \delta_2(\infty, f)+ \min\Biggl\{\sum_{a\neq 0,1,\infty} \delta_2(a, f),\sum_{a\neq 0,1,\infty} \delta_2(a, g)\Biggr\}> {5m-1\over m-1}, \] then either \(f\equiv g\) or \(fg\equiv 1\).
Reviewer: Yu Jiarong (Wuhan)

MSC:

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