Drasin's problems and Mues' conjecture (Q1206067)

Several results concerning the deficiencies of meromorphic functions are obtained. Denote, as usual, the deficiency of \(f\) at \(a\) by \(\delta(a,f)\). It is proved that if \(k\geq 1\), then \[ \sum_{a\in C} \delta(a,f)+\sum_{b\in C} \delta(b,f^{(k)})\leq 3. \] This answers a question of D. Drasin. The inequality is sharp and further results are given in the case of equality. Another result obtained in this paper is that there exists a constant \(K(f)\) such that if \(k\geq K(f)\), then \[ \sum_{a\in C} \delta(a,f^{(k)})\leq 1.\tag{1} \] This is a partial solution to a conjecture of E. Mues which says that (1) holds for all \(k\geq 1\). In a forthcoming paper, the second author has improved on the latter result by showing that there are at most four values of \(k\) for which (1) can fail.