Nonlinear differentiation equation and analytic function spaces

Abstract

In this paper we consider the nonlinear complex differential equation $$(f^{(k)}{)}^{n_k}+A_{k-1}(z)(f^{(k-1)}{)}^{n_{k-1}}+\cdot \cdot \cdot +A_1(z)(f^{\prime }{)}^{n_1}+A_0(z)f^{n_0}=0,$$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 Möbius 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.