A note of spirallike mappings in \({\mathbb{C}}^n\) (Q2716404)

Let \(\mathbb B^n\subset\mathbb C^n\) be the unit ball, and \(f\) a normalized biholomorphic mapping in \(\mathbb B^n.\) Let \(A\in L(\mathbb C^n)\) be a normal operator for which \(\Re\lambda>0\) for each eigenvalue \(\lambda\) of \(A\). Then \(f\) is spirallike relative to \(A\) if \(e^{At}f(\mathbb B^n)\subset f(\mathbb B^n)\) for all \(t\leq 0\). NEWLINENEWLINENEWLINEThe main results of the paper are the growth, and Koebe-1/4 theorems: If \(f:\mathbb B^n\to \mathbb C^n\) is a spirallike mapping relative to \(A\), then for any \(z\in \mathbb B^n\) the inequality NEWLINE\[NEWLINE\|z \|/(1 + \|z \|)^2\leq \|f(z)\|\leq \|z \|/(1- \|z \|)^2NEWLINE\]NEWLINE holds. These estimates are precise. As a consequence \(f(\mathbb{B}^n)\supset\frac{1}{4}\mathbb{B}^n.\)