Relations between solutions of differential equations and small functions
linear differential equation in \(\mathbb{C}\)
In this paper, the authors use Nevanlinna theory and oscillation theory of meromorphic functions to study the higher order linear differential equations NEWLINE\[NEWLINE f^{(k)}+A_{k-1}(z)f^{(k-1)}+\dots+A_{1}(z)f'+A_{0}(z)f=0, NEWLINE\]NEWLINE where \(A_{k-1}(z),\dots,A_{1}(z)\) and \(A_{0}(z)\) are entire functions of finite order such that if \(A_{j}(z)\not\equiv 0\), then \(\lambda(A_{j})<\sigma(A_{j})\); if \(i\neq j\), then \(\sigma(A_{i}/A_{j})=\max\{\sigma(A_{i}), \sigma(A_{j})\}\). The authors derive some relations between the meromorphic solutions of such an equation and prove that every meromorphic solution and its derivative have infinitely many fixed points.
scientific article; zbMATH DE number 6034207
scientific article