On some sufficient conditions for starlikeness of order \(\alpha\) in \(\mathbb{C}^n\)
type: Item
instance of: scholarly article
title: On some sufficient conditions for starlikeness of order \(\alpha\) in \(\mathbb{C}^n\)
zbMATH Open document ID: 1117.30012
author: Ming-Sheng Liu
author: Yucan Zhu
published in: Taiwanese Journal of Mathematics
publication date: 2007-01-29T00:00:00Z
review text: The authors use certain differential inequalities to obtain sufficient conditions for starlikeness of order \(\alpha\) on the unit ball in \(\mathbb{C}^n\). Let \(B^n\) be the unit ball in \(\mathbb{C}^n\) with the Euclidean inner product \(\langle\cdot,\cdot\rangle\) and the norm \(\| z\| =\sqrt{\langle z,z\rangle}\). A locally biholomorphic mapping \(f\) on \(B^n\), such that \(f(0)=0\) and \(Df(0)=I_n\) (\(I_n\) is the \(n\times n\) unit matrix), is called starlike of order \(\alpha\in (0,1)\) if \[ \left| \frac{1}{\| z\| ^2}\langle [Df(z)]^{-1}f(z),z\rangle-\frac{1}{2\alpha}\right| <\frac{1}{2\alpha} \] for \(z\in B^n\setminus\{0\}\). In Theorem 1, the authors obtain a sufficient condition of starlikeness of order \(\alpha\) on \(B^n\), which in the case of one complex variable generalizes a result due to P. T. Mocanu and G. Oros. Further, the authors obtain a distortion theorem and a covering result.
reviewed by: G. Kohr
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5119516
zbMATH Keywords: biholomorphic mapping
zbMATH Keywords: covering theorem
zbMATH Keywords: local biholomorphic mapping
zbMATH Keywords: starlike mapping
zbMATH Keywords: starlikeness of order \(\alpha\)
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500557296
OpenAlex ID: W4239735676
DOI: 10.11650/TWJM/1500557296
description: scientific article; zbMATH DE number 5119516