On the univalence of polyanalytic functions

Abstract

A continuous complex-valued function $F$ in a domain $D\subseteq \mathbf{C}$ is Poly-analytic of order $\alpha$ if it satisfies ${\partial }_{\overline{z}}^{\alpha }F=0.$ One can show that $F$ has the form $F(z)=\sum _0^{n-1}{\overline{z}}^k{A}_k(z)$, where each $A_k$ is an analytic function$.$ In this paper, we prove the existence of a Landau constant for Poly-analytic functions and the special Bi-analytic case. We also establish the Bohr's inequality for poly-analytic and bi-analytic functions which map $U$ into $U$. In addition, we give an estimate for the arclength over the class of poly-analytic mappings and consider the problem of minimizing moments of order $p$.