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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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##### References:
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