Spirallike mappings on bounded balanced pseudo-convex domains in \(\mathbb{C}^n\) (Q2782690)

Let \(\Omega\subset\mathbb C^n\) be a bounded balanced pseudoconvex domain with \(C^1\) plurisubharmonic defining functions, and let \(f:\Omega\to \mathbb C^n\) be a locally biholomorphic mapping normalized by \(f(0) = 0\) and \(Df(0) = I\). NEWLINENEWLINENEWLINEIn this paper, some conditions of spirallikeness are given: The mapping \(f\) is spirallike iff \(f\) is biholomorphic and there exist \(\lambda_j\), with \(\Re \lambda_j > 0\), and a unitary transformation \(V\) such that each of the coordinate projections \((Vf)_j(\Omega):\Omega\to\mathbb C\) contains all of the spirals \(\{(Vf)(z)_j\exp\{-\lambda_jt\}: t\geq 0\}\), \(z\in\Omega\), \(j=1,\dots,n.\) Let \(\alpha\in\mathbb R\) with \(|a|<\pi/2\) and assume that NEWLINE\[NEWLINE\Re\left[e^{-ia}\left\langle\left[Df(z)\right]^{-1}f(z),\frac{\partial h^2}{\partial\overline z}(z)\right\rangle\right]> 0,\quad z\in\Omega\setminus\{0\}.NEWLINE\]NEWLINE Then \(f\) is spirallike, hence univalent on \(\Omega\).