The nonzero solutions to a class of linear differential equations (Q2748861)

The author uses the Nevanlinna theory to study the value distribution of entire solutions and its derivatives to the linear equation NEWLINE\[NEWLINEw^{(k)}+A_{k-1}w^{(k-1)}+\cdots +A_1w^\prime+(A_0+A)w=0,\tag{1}NEWLINE\]NEWLINE where \(A_j, j=0, \cdots, k-1\), and \(A\) are entire functions, and \(A\) is nonconstant and such that \(T(A_j, r)=o(T(r, A))\) for \(j=0, \cdots, k-1\). The main result proves that, for any nontrivial solution \(f\) to (1) and with \(n\in \mathbb{N}\), \(f\) satisfies NEWLINENEWLINENEWLINE(i) \(N(r, 1/f)=N(r, 1/f^{(n)})+S(r, f)\), NEWLINENEWLINENEWLINE(ii) \(\delta(0, f)=\delta(0, f^{(n)})\) when \(f\) has finite order, andNEWLINENEWLINENEWLINE(iii) \(\delta(c, f)=\delta(c, f^{(n)})=0\) for any \(T(r, c)=S(r, A)\). NEWLINENEWLINENEWLINEThe proof is based on standard results from the Nevanlinna theory. The author mentions that the main result of her paper generalizes an earlier result in a preprint of X. H. Hau and Y. Lin concerning second-order equations of (1), but the reviewer was unable to locate the paper. The main result should be compared with results due to \textit{S. Bank, G. Frank} and \textit{I. Laine} [Math. Z. 183, 355-364 (1983; Zbl 0506.34005)]. The main result is also closely related to the complex oscillation theory of \textit{I. Laine}[Nevanlinna theory and complex differential equations (Berlin: Walter de Gruyter) (1992; Zbl 0784.30002)] and \textit{S. Gao} [Proc. Edinb. Math. Soc., II. Ser. 43, No. 1, 1-13 (2000; Zbl 0956.34076)]. The use of notations for the coefficients in equations (5) and (8) is confusing. There are also a number of typing mistakes in the paper.