Weighted sharing of three values
The following definitions are known. Meromorphic functions \(f\) and \(g\) share the value \(a\) CM (counting multiplicities) if \(f\) and \(g\) have the same \(a\)-points with the same multiplicities. If the multiplicities are ignored then functions \(f\) and \(g\) share the value \(a\) IM. Let \(k\in \{ 0,1,2, \ldots , \infty \}\), \(a\in {\bar\mathbf{C}},\) and let \(f\) be a meromorphic function. The set of all \(a\)-points of \(f\) where an \(a\)-point of multiplicity \(m\) is counted \(m\) times if \(m \leq k\) and \(k+1\) times if \(m>k\) is denoted by \(E_k(a,f).\) Earlier the author generalizing the previous definitions introduced the following. Meromorphic functions \(f\) and \(g\) share a value \(a\) with weight \(k\) if \(E_k(a,f)= E_k(a,g).\) The author makes use of value analogic to Nevanlinna deficiency \[ \delta_{s)}(a,f) = 1 - \limsup_{r \to \infty} \frac{N(r,a,f| \leq s)}{T(r,f)} , \] where \(N(r,a,f| \leq s)\) is the counting function of those \(a\)-points of \(f\) whose multiplicities are less then or equal to \(s\) and \(a\)-points are counting according to its multiplicities. The author proves three theorems. Theorem 1. Let \(f\) and \(g\) be two distinct nonconstant meromorphic functions sharing \((0,1)\), \((1,m)\), \((\infty , k),\) where \((m-1)(mk-1) > (m+1)^2.\) If \(a (\neq 0, 1, \infty )\) is a complex number such that \(3\delta_{2)}(a,f) + 2\delta_{1)}(\infty , f) > 3,\) then \(a\) and \(1-a\) are Picard exceptional values of \(f\) and \(g\) respectively and also \(\infty\) is a Picard exceptional value of both \(f\) and \(g\) and \((f-a)(g+a-1) \equiv a(1-a).\) Theorems 2 and 3 have the same conclusion under other conditions. These theorems develop the theorem of S.Z. Ye and the earlier theorems of the author.
scientific article; zbMATH DE number 2112818