Sharing values and a problem due to C. C. Yang (Q1358944)

Let \(f,g,a\) and \(b\) be meromorphic functions in the plane such that \(T(r,a)= S(r,f)\) and \(T(r,b)= S(r,g)\). If the two functions \(f-a\) and \(g-b\) have the same zeros with the same multiplicities, then one says: \(f\) and \(g\) share the pair \((a,b)CM\). In the paper under review the author proves: Let \(f,g,a,\alpha,b\) and \(\beta\) be meromorphic functions in the plane, where \(a\) and \(\alpha\) are small functions with respect to \(f\) and \(b\) and \(\beta\) are small functions with respect to \(g\), i.e. \(T(r,a)\), \(T(r,\alpha) =S(r,f)\) and \(T(r,b)\), \(T(r,\beta) =S(r,g)\) and \(a\neq \alpha\) and \(b\neq\beta\). Suppose that \(f\) and \(g\) share the pair \((a,b)CM\) and \(\delta (\alpha,f) +\delta(\beta,g) +\delta(\infty,f) +\delta (\infty,g)>3\). Then either \[ {f-\alpha \over a- \alpha}= {g-\beta \over b- \beta} \quad\text{or} \quad {f-\alpha \over a-\alpha} \cdot {g-\beta \over b-\beta} =1. \] He also shows by the following example that the number 3 in the above inequality is best possible: \[ f(z)= e^{2z}- Q(z)e^z, \quad g(z)=e^{2z}/ \left(e^z+ {P(z) \over Q(z)} \right), \] where \(P\) and \(Q\) are nonzero polynomials. Here \(f\) and \(g\) share the pair \((P,Q)CM\) and \(\alpha= \beta=0\).