Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations"

Abstract

Let $\phi$, $\psi \in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. Suppose that $f\in C^4(\mathbb{D})$ satisfies the following: (1) the inhomogeneous biharmonic equation $\Delta (\Delta f(z))=g(z)$ for $z\in \mathbb{D}$, (2) the Dirichlet boundary conditions ${\partial }_{\overline{z}}f(\zeta )=\phi (\zeta )$ and $f(\zeta )=\psi (\zeta )$ for $\zeta \in \mathbb{T}$. Recently, the authors in [J. Geom. Anal. 29: 2469-2491, 2019] showed that if $\omega$ is a majorant with ${\mathrm{limsup}}_{t\to 0^{+}}\left(\omega (t)/t\right)<\infty$, $\psi =0$ and ${\phi }_1\in {\mathcal{L}}_{\omega }(\mathbb{T})$, where ${\phi }_1(e^{it})=\phi (e^{it})e^{-it}$ for $t\in [0,2\pi ]$, then $f\in {\mathcal{L}}_{\omega }(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "${\mathrm{limsup}}_{t\to 0^{+}}\left(\omega (t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$\psi =0$" and "${\phi }_1\in {\mathcal{L}}_{\omega }(\mathbb{T})$" can be replaced by weaker conditions.

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