On the growth of solutions of \(w^{(n)}+e^{-z}w'+Q(z) w=0\) and some related extensions (Q2641677)

The differential equation \[ w''+e^{-z}w'+B(z)w=0, \tag{1} \] where \(B(z)\) is an entire function, was studied by \textit{M. Frei} [Comment. Math. Helv. 36, 1--8 (1961; Zbl 0115.06904)], \textit{M. Ozawa} [Kodai. Math. J. 3, 295--309 (1980; Zbl 0463.34028)], \textit{G. G. Gundersen} [Proc. R. Soc. Edinb. A 102, 9--17 (1986; Zbl 0598.34002)] and \textit{J. K. Langley} [Kodai Math. J. 9, 430--439 (1986; Zbl 0609.34041)]. In particular Langley proved that if \(B(z)\) is a nonconstant polynomial, then every solution \(w\not\equiv 0\) of (1) has infinite order. In this paper, the authors try to generalize these results to the differential equation \[ w^{(n)}+e^{-z}w'+B(z)w=0. \tag{2} \] They prove that if the differential equation \[ w^{(n)}+e^{-z}w'+cw=0, \tag{3} \] where \(c\not=0\) is a complex constant, has a solution \(w\not\equiv 0\) of finite order, then \(c=-k^n,\) where \(k\) is some positive integer. Conversely, for each positive integer \(k\), the above equation with \(c=-k^n\) has a solution of the type \(w=P(e^z),\) where \(P\) is a polynomial of degree \(k.\) The authors claim that if \(B(z)\) is a nonconstant polynomial, then every nonzero solution of the differential equation (2) with \(n\geq 2\) has infinite order. However, there exists a gap in the proof. We can not obtain \(| Q(z)| < M\) from (4.6) in the proof of Theorem 1.2 on page 579, where \(M\) is a positive constant which is independent of \(z.\) We can only obtain that \(| Q(z)| < Me^{| z| \rho\cos\epsilon}\) from (4.6), where \(M\) is a positive constant which is independent of \(z.\) This is not enough to get the conclusion. There also exists a gap in the proof of Theorem 1.3