Weighted sharing and uniqueness of meromorphic functions (Q2725476)

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.