Weighted sharing and uniqueness of meromorphic functions. (English) Zbl 0981.30023

Lahiri, Indrajit

Nagoya Math. J. 161, 193-206 (2001).

This article proposes an idea of weighted shared values for meromorphic functions, resulting in improvements of some previous shared value results. As to the definition, given \(k\in\mathbb N_0\cup\{\infty\}\) and \(a\in\mathbb C\cup\{\infty\}\), let \(E_k(a;f)\) denote the set of all \(a\)-points of \(f\), counting an \(a\)-point according to its multiplicity \(m\), if \(m\leqq k\) and \(k+1\) times, if \(m>k\). If now \(E_k(a;f)=E_k(g;f)\), we say that \(f,g\) share \((a,k)\). Clearly, sharing \((a,0)\), resp.\((a,\infty)\), equals to sharing a \(IM\), resp.\(CM\). Denoting now by \(N(r,a;f|=1)\) the integrated function for simple \(a\)-points of \(f\), it is well-known, see [H.-X. Yi, Kodai Math. J. 13, No. 3, 363-372 (1990; Zbl 0712.30029)], that if \(f,g\) share \(0,1\) and \(\infty\) \(CM\) and if \(N(r,0;f|=1)+N(r,\infty;f|=1)<\{\lambda+o(1)\}\max(T(r,f),T(r,g))\), where \(0<\lambda<1/2\), in a set of \(r\)-values of infinite linear measure, then either \(f=g\) or \(fg=1\). The improvement now proves the same conclusion, provided \(f,g\) share \((0,1)\), \((\infty,\infty)\) and \((1,\infty)\). The conclusion also follows whenever \(f,g\) share \((0,1)\), \((\infty,0)\) and \((1,\infty)\) and \(N(r,0;f|=1)+4\bar{N}(r,\infty;f)<\{\lambda+o(1)\}\max(T(r,f),T(r,g))\). The proofs apply careful considerations with the Nevanlinna theory. The paper is clearly written, including some illuminating examples.

Reviewer: Ilpo Laine (Joensuu)

MSC:

30D35

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

Citations:

Zbl 0712.30029

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