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On some results of Lahiri. (English) Zbl 1045.30018
Let \(f(z)\) and \(g(z)\) be transcendental meromorphic functions in the complex plane. The author considers uniqueness problems of meromorphic functions with weighted sharing value introduced by I. Lahiri, see e.g. [Kodai Math. J. 24, No. 3, 421–435 (2001; Zbl 0996.30020)]. A value \(a\in\widehat{\mathbb C}= \mathbb C\cup\{\infty\}\) is called an IM (ignoring multiplicities) shared value of \(f(z)\) and \(g(z)\), \(f(z)= a\) if and only if \(g(z)= a\). A value \(a\in \widehat{\mathbb C}\) is called a CM (counting multiplicities) shared value of \(f(z)\) and \(g(z)\) if \(f(z)\) and \(g(z)\) assume \(a\) at the same points with the same multiplicities. Let \(k\) be a positive integer. Denote by \(E_k(a,f)\) the set of all \(a\)-points of \(f(z)\) of multiplicity \(m\) is counted \(m\) times if \(m\leq k\) and \(k+1\) times if \(m> k\). If \(E_k(a, f)= E_k(a, g)\), then it is said that \(f(z)\) and \(g(z)\) share the value \(a\) with weighted \(k\). Regard the case \(f(z)\) and \(g(z)\) share \(a\)-points CM to \(k=\infty\) in the definition above. The author writes \(f(z)\) and \(g(z)\) share \((a,k)\) to mean that \(f(z)\) and \(g(z)\) share \(a\) with weighted \(k\). Denote by \(N_{k)}(r, 1/(f-a))\) the counting function of \(a\)-points of \(f(z)\) of multiplicity at most \(k\). The author shows that if \(f(z)\) and \(g(z)\) share \((0,1)\), \((\infty,0)\) and \((1,5)\) and if for some interval \(I\in \mathbb R\) \[ \lim_{r\to\infty}\, \sup_{r\in I}\;{N_{1)}(r, 1/f)+ 3\overline N(r,f)- (1/2)m(r, 1/(g- 1))\over T(r,f)}< \frac12, \] then either \(f(z)\equiv g(z)\) or \(f(z) g(z)\equiv 1\). This result is an improvement of the result due to I. Lahiri [Nagoya Math. J. 161, 193–206 (2001; Zbl 0981.30023)].

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