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: (∂z∂z¯)2w=g in the unit disc D, w= γ and ∂ν∂z∂z¯w= γ on T (the unit circle), 12πi∫Twζζ¯(ζ)dζζ=c, where ∂ν 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.