Nonlinear differentiation equation and analytic function spaces

Abstract: In this paper we consider the nonlinear complex differential equation $$(f^{(k)})^{n_{k}}+A_{k-1}(z)(f^{(k-1)})^{n_{k-1}}+\cdot\cdot\cdot+A_{1}(z)(f')^{n_{1}}+A_{0}(z)f^{n_{0}}=0, $$where $ A_{j}(z)$, $ j=0, \cdots, k-1 $, are analytic in the unit disk $ \mathbb{D} $, $ n_{j}\in R^{+} $ for all $ j=0, \cdots, k $. We investigate this nonlinear differential equation from two aspects. On one hand, we provide some sufficient conditions on coefficients such that all solutions of this equation belong to a class of M\"{o}bius invariant function space, the so-called $Q_K$ space. On the other hand, we find some growth estimates for the analytic solutions of this equation if the coefficients belong to some analytic function spaces.