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On the Univalence of Poly-analytic Functions
Computational Methods and Function Theory ( IF 2.1 ) Pub Date : 2021-04-27 , DOI: 10.1007/s40315-021-00378-5
Zayid Abdulhadi , Layan El Hajj

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.



中文翻译:

关于多元解析函数的等价性

连续复值函数˚F域中\(d \ subseteq \ mathbf {C} \)是聚解析的顺序\(\阿尔法\) ,如果它满足\(\局部^ {\阿尔法} F / \局部\ overline {z} ^ {\ alpha} = 0 \)。可以证明F的形式为\(F(z)= \ sum _ {k = 0} ^ {\ alpha -1} \ overline {z} ^ {k} A_ {k}(z)\),其中每个\(A_k \)是一种分析功能。在本文中,我们证明了多元分析函数和特殊双解析情况的Landau常数的存在。我们还建立了多元分析和双向分析函数的玻尔不等式。另外,我们对多解析映射类上的弧长进行了估计,并考虑了最小化阶数p的问题

更新日期:2021-04-28
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