Schwarz Lemma for mappings satisfying Biharmonic Equations

Abstract

In this paper, we establish some Schwarz type lemmas for mappings $\Phi$ satisfying the inhomogeneous biharmonic Dirichlet problem $\Delta (\Delta (\Phi ))=g$ in $\mathbb{D}$, $\Phi =f$ on $\mathbb{T}$ and ${\partial }_n\Phi =h$ on $\mathbb{T}$, where $g$ is a continuous function on $\overline{\mathbb{D}}$, $f,h$ are continuous functions on $\mathbb{T}$, where $\mathbb{D}$ is the unit disc of the complex plane $\mathbb{C}$ and $\mathbb{T}=\partial \mathbb{D}$ is the unit circle. To reach our aim, we start by investigating some properties of $T_2$-harmonic functions. Finally, we prove a Landau-type theorem.

arXiv admin note: text overlap with arXiv:1801.04507 by other authors