Schwarz-Pick and Landau type theorems for solutions to the Dirichlet-Neumann problem in the unit disk
Abstract
The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $(\partial_z\partial_{\overline{z}})^2 w=g$ in the unit disc $\ID$, $w=\gamma_0$ and $\partial_{\nu}\partial_z\partial_{\overline{z}}w=\gamma$ on $\mathbb{T}$ (the unit circle), $\frac{1}{2\pi i}\int_{\mathbb{T}}w_{\zeta\overline{\zeta}}(\zeta)\frac{d\zeta}{\zeta}=c$, where $\partial_\nu$ denotes differentiation in the outward normal direction. More precisely, we obtain Schwarz-Pick type inequalities and Landau type theorem for solutions to the Dirichlet-Neumann problem.
- Publication:
-
arXiv e-prints
- Pub Date:
- March 2021
- DOI:
- 10.48550/arXiv.2103.09112
- arXiv:
- arXiv:2103.09112
- Bibcode:
- 2021arXiv210309112L
- Keywords:
-
- Mathematics - Complex Variables;
- Primary: 30C62;
- 31A30;
- 30C80;
- Secondary: 30C20;
- 31A05
- E-Print:
- 16 pages