On the Univalence of Poly-analytic Functions

A continuous complex-valued function F in a domain D ⊆ C is poly-analytic of order α if it satisfies ∂ α F / ∂ z ¯ α = 0 . One can show that F has the form F ( z ) = ∑ k = 0 α - 1 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 .