On the Univalence of Poly-analytic Functions

Abstract

A continuous complex-valued function F in a domain $$D\subseteq \mathbf {C}$$ is poly-analytic of order $$\alpha $$ if it satisfies $$\partial ^{\alpha }F/\partial \overline{z}^{\alpha }=0$$ . One can show that F has the form $$F(z)=\sum _{k=0}^{\alpha -1}\overline{z}^{k}A_{k}(z)$$ , 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 Bohr’s inequality for poly-analytic and bi-analytic functions. In addition, we give an estimate for the arc-length over the class of poly-analytic mappings and consider the problem of minimizing moments of order p.