Normal criteria of function families related to a result of Drasin (Q659545)

Summary: We study the normality of families of meromorphic functions related to a result of \textit{D. Drasin} [Acta Math. 122, 231--263 (1969; Zbl 0176.02802 )]. We consider whether a family \(\mathcal F\) of meromorphic functions without zeros is normal in \(D\), if for every pair of functions \(f\) and \(g\) in \(\mathcal F\), \(f\) and \(g\) share \(\infty\) or \(H(f) - 1\) and \(H(g) - 1\) share 0, where \(H(f) := f^{(k)}(z) + a_{k-1}f^{(k-1)}(z) + \cdots + a_0f(z)\). Some examples show that the conditions in our results are best possible.