Schwarz type lemma, Landau type theorem and Lipschitz type space of solutions to biharmonic equations

TL;DR

This paper investigates properties of solutions to biharmonic equations in the complex plane, establishing Schwarz and Landau type lemmas and analyzing Lipschitz continuity to deepen understanding of their behavior.

Contribution

It introduces new Schwarz and Landau type theorems and explores Lipschitz properties for solutions to biharmonic equations, advancing theoretical understanding.

Findings

Established Schwarz type lemma for biharmonic solutions

Derived a Landau type theorem based on the results

Analyzed Lipschitz continuity of solutions

Abstract

The purpose of this paper is to study the properties of the solutions to the biharmonic equations: $\Delta(\Delta f)=g$, where $g:$ $\overline{\mathbb{D}}\rightarrow\mathbb{C}$ is a continuous function and $\overline{\mathbb{D}}$ denotes the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$. In fact, we establish the following properties for those solutions: Firstly, we establish the Schwarz type lemma. Secondly, by using the obtained results, we get a Landau type theorem. Thirdly, we discuss their Lipschitz type property.