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.

本文旨在建立Dirichlet–Neumann问题解的一些性质：\（（\ partial _z \ partial _ {\ overline {z}}）^ 2 w = g \）在单位圆盘\（{ {\ mathbb {d}}} \），\（W = \伽马_0 \）和\（\局部_ {\ NU} \局部_z \局部_ {\划线{Z}} w = \伽马\）上\（{\ mathbb {T}} \）（单位圆），\（\ frac {1} {2 \ pi i} \ int _ {{\ mathbb {T}}} w _ {\ zeta {\ overline { \ zeta}}}（\ zeta）\ frac {d \ zeta} {\ zeta} = c \），其中\（\ partial _ \ nu \）表示在向外法线方向上的微分。更准确地说，我们获得了Dirichlet-Neumann问题解的Schwarz-Pick型不等式和Landau型定理。