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 ${{\mathbb {D}}}$, $w=\gamma _0$ 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.