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On the Univalence of Poly-analytic Functions

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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.

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Authors and Affiliations

  1. Department of Mathematics, American University of Sharjah, Box 26666, Sharjah, United Arab Emirates

    Zayid Abdulhadi

  2. Department of Mathematics, American University of Dubai, Box 28282, Dubai, United Arab Emirates

    Layan El Hajj

Authors
  1. Zayid Abdulhadi
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  2. Layan El Hajj
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Correspondence to Layan El Hajj.

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Communicated by Pekka Koskela.

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Abdulhadi, Z., Hajj, L.E. On the Univalence of Poly-analytic Functions. Comput. Methods Funct. Theory 22, 169–181 (2022). https://doi.org/10.1007/s40315-021-00378-5

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  • DOI: https://doi.org/10.1007/s40315-021-00378-5

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