Exceptional sets related to Hayman's alternative (Q864694)

\textit{W. K. Hayman} [Meromorphic functions. Oxford: Clarendon Press (1964; Zbl 0115.06203)] proved for \(f\) a transcendental meromorphic function in the complex plane \(\mathbb{C}\) that either \(f\) assumes every finite value infinitely often or each of its derivatives \(f^{(k)}\), \(k\geq 1\), assumes every finite non-zero value infinitely often. If \(k\) is a positive integer and \(E\) is a countable subset of \(\mathbb C\) with points tending to \(\infty\), which also satisfies a spacing condition depending on \(k\), then for a meromorphic function \(f\) in \(\mathbb C\) which has sufficiently large (depending on \(k\)) Nevanlinna deficiency at the poles of \(f\), either \(f\) assumes every finite value infinitely often in \(\mathbb C\setminus E\), or \(f^{(k)}\) takes every non-zero complex value infinitely often in \(\mathbb C\setminus E\). The theorem complements results of \textit{J. K. Langley} [Ann. Acad. Sci. Fenn., Ser. A I Math. 11, 137--153 (1986; Zbl 0563.30023)].