an:01619257 Zbl 0981.30023 Lahiri, Indrajit Weighted sharing and uniqueness of meromorphic functions EN Nagoya Math. J. 161, 193-206 (2001). 00073026 2001
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30D35 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 [\textit{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. Ilpo Laine (Joensuu) Zbl 0712.30029