A remark on Langley's generalisation of Hayman's alternative (Q1320075)

Let \(f\) be transcendental meromorphic in the plane and set \(\psi= f^{(k)}+ a_{k-1} f^{(k-1)}+\cdots + a_ 0 f\), \(k\geq 1\), where \(a_ 0,\dots, a_{k-1}\) are small functions with respect to \(f\) (i.e.,\(T(r,a_ j)= S(r,f)\)). For \(\psi= f^{(k)}- 1\) \textit{W. K. Hayman} [Ann. Math., II. Ser. 79, 9-42 (1959; Zbl 0088.285)] proved the estimate \[ T(r,f)\leq 3N\Bigl(r,\textstyle{{1\over f}}\Bigr)+ 4\overline N\Bigl(r,\textstyle{{1\over \psi}}\Bigr)+ S(r,f). \] \textit{J. K. Langley} [Math. Z. 187, 1-11 (1984; Zbl 0563.30024), Mathematika 32, 132-146 (1985; Zbl 0559.30035)] has shown that this remains true for general \(\psi\), if some exceptional case is avoided. In the present paper, it is shown that in the exceptional case \(f\) has the form \({(H- \omega)^{k+1}\over k!H(H')^ k}\), where \(\omega^{k+1}= 1\) and \(H\) is meromorphic satisfying \({H''\over H'}= {2a_{k-1}\over k(k+ 1)}\).