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Remarks on "Schwarz-type lemma, Landau-type theorem, and Lipschitz-type space of solutions to inhomogeneous biharmonic equations"
Complex Variables · Submitted 18 Mar 2025
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$\psi=0$" and "$\varphi_1\in\mathscr{L}_{\omega}(\mathbb{T})$" can be replaced by weaker conditions.
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Let $\varphi$, $\psi\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, 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 $\partial_{\overline{z}}f(\zeta)=\varphi(\zeta)$ 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 $\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$, $\psi=0$ and $\varphi_1 \in\mathscr{L}_{\omega}(\mathbb{T})$, where $\varphi_1(e^{it})=\varphi(e^{it})e^{-it}$ for $t\in[0,2\pi]$, then $f\in\mathscr{L}_{\omega}(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$\psi=0$" and "$\varphi_1\in\mathscr{L}_{\omega}(\mathbb{T})$" can be replaced by weaker conditions.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $\varphi$, $\psi\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, where $\mathbb{D}$ and $\mathbb{T}$ denote the unit disk and the unit circle, respectively. | background 1.00 | 0.10 | 0.01 | asserted 0.52 |
| s1 | 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 $\partial_{\overline{z}}f(\zeta)=\varphi(\zeta)$ and $f(\zeta)=\psi(\zeta)$ for $\zeta\in\mathbb{T}$. | background 0.99 | 0.11 | 0.02 | asserted 0.72 |
| s2 | Recently, the authors in [J. Geom. | background 1.00 | 0.03 | 0.01 | unclear 0.89 |
| s3 | Anal. 29: 2469-2491, 2019] showed that if $\omega$ is a majorant with $\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$, $\psi=0$ and $\varphi_1 \in\mathscr{L}_{\omega}(\mathbb{T})$, where $\varphi_1(e^{it})=\varphi(e^{it})e^{-it}$ for $t\in[0,2\pi]$, then $f\in\mathscr{L}_{\omega}(\mathbb{D})$. | background 0.99 | 0.03 | 0.07 | asserted 1.00 |
| s4 | The purpose of this paper is to improve and generalize this result. | contribution 0.97 | 0.22 | 0.07 | asserted 0.86 |
| s5 | We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$\psi=0$" and "$\varphi_1\in\mathscr{L}_{\omega}(\mathbb{T})$" can be replaced by weaker conditions. | contribution 0.91 | 0.94 | 0.81 | asserted 1.00 |
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Original abstract
Let $\varphi$, $\psi\in C(\mathbb{T})$, $g\in C(\overline{\mathbb{D}})$, 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 $\partial_{\overline{z}}f(\zeta)=\varphi(\zeta)$ 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 $\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$, $\psi=0$ and $\varphi_1 \in\mathscr{L}_{\omega}(\mathbb{T})$, where $\varphi_1(e^{it})=\varphi(e^{it})e^{-it}$ for $t\in[0,2\pi]$, then $f\in\mathscr{L}_{\omega}(\mathbb{D})$. The purpose of this paper is to improve and generalize this result. We not only prove that the condition "$\limsup_{t\rightarrow0^{+}}\left(\omega(t)/t\right)<\infty$" is redundant, but also demonstrate that conditions "$\psi=0$" and "$\varphi_1\in\mathscr{L}_{\omega}(\mathbb{T})$" can be replaced by weaker conditions.
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