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.

16 pages; This is to appear in Computational Methods and Function Theory