On some sufficient conditions for starlikeness of order \(\alpha\) in \(\mathbb{C}^n\)
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.
scientific article; zbMATH DE number 5119516