Schwarz-Pick and Landau type theorems for solutions to the Dirichlet-Neumann problem in the unit disk

The aim of this paper is to establish some properties of solutions to the Dirichlet-Neumann problem: $({\mathrm{\partial }}_z{\mathrm{\partial }}_{\bar{z}}{)}^{2}w=g$ in the unit disc $\text{ID}$, $w={\gamma }_0$ and ${\mathrm{\partial }}_{\nu }{\mathrm{\partial }}_z{\mathrm{\partial }}_{\bar{z}}w=\gamma$ 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 ${\mathrm{\partial }}_{\nu }$ 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.