A note on Mues' conjecture (Q1348720)

Let \(f\) denote a transcendental meromorphic function in \(\mathbb{C}\). \textit{E. Mues} posed the following conjecture [ Manucr. Math. 5, 275--297 (1971; Zbl 0225.30031)] for any positive integer \(k\), (1) \(\sum_{a\neq\infty} \delta (a,f^{(k)}\leq 1\), where \(\delta(a,f^{(k)})\) denotes the deficiency in the sense of Nevanlinna. Mues himself showed in the same paper that under some conditions on \(f\), (1) is true. Later, the reviewer in Arch. Math. 55, 374--379 (1990; Zbl 0698.30029)] \textit{L. Yang} and \textit{Y. Wang} in Sci. China, Ser. A 35, 1180--1190 (1992; Zbl 0760.30011)] found improvements of the Mues result. However, so far as I know, this conjecture has not been solved in most general case. In the present paper, the author obtained a partial result to the Mues conjecture. If \(N(r,\alpha, f\mid=1)= S(r,f)\) for some \(\alpha\neq\infty\), then (1) holds for \(k\geq 2\). This theorem implies that if \(f\) can be written as \(f= \alpha +g^m\), where \(\alpha\in \mathbb{C}\), \(m\geq 2\) is an integer and \(g\) is a meromorphic function, then (1) holds for such \(f\). The methods in the proof strongly depend on the Nevanlinna second fundamental theorem, the Hayman-Miles theorem and the inequality that is obtained due to Wang and Yang.