Weighted sharing three values and Brosch's theorem
Two meromorphic functions \(f\) and \(g\) are said to share a value \(a\) with weight \(k\), if they have the same \(a\)-points and if the multiplicity of these \(a\)-points is equal whenever it is less than or equal to \(k.\) Suppose that \(f\) and \(g\) share one of the values \(0,1,\infty\) with weight 1 while the two others are shared with weight \(\infty.\) Let \(N_0(r)\) be the counting function of the zeros of \(f-g\) which are not zeros of \(f,f-1\) or \(1/f\) and suppose that \(N_0(r)\neq S(r,f).\) It is shown that then all three values \(0,1,\infty\) are shared with weight \(\infty.\) A second result says that if the condition on \(N_0(r)\) is replaced by a condition on the counting functions of certain points, then \(f\) and \(g\) are of a specific form. (The list of possible forms contains 10 cases if \(f\) is a linear transformation of \(g\) and 9 cases otherwise.)
scientific article; zbMATH DE number 5072925