Nonlinear differentiation equation and analytic function spaces

TL;DR

This paper studies a nonlinear complex differential equation, providing conditions for solutions to belong to specific function spaces and establishing growth estimates based on coefficient properties.

Contribution

It introduces new criteria for solutions to be in $Q_K$ spaces and derives growth bounds depending on the coefficients' analytic function spaces.

Findings

Solutions belong to $Q_K$ spaces under certain coefficient conditions

Growth estimates are established for solutions based on coefficient spaces

Provides a link between differential equation solutions and analytic function space properties

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.