A starlikeness criterion for holomorphic mappings on balanced pseudoconvex domains (Q2782683)

The author gives the following starlikeness criterion: Let \(\Omega\subset\mathbb C^n\) be a bounded balanced domain with \(C^1\) plurisubharmonic defining functions, \(h\) be the Minkowski function of \(\Omega\), and \(f\) be a locally biholomorphic mapping on \(\Omega\) satisfying \(f(0) = 0\) and NEWLINE\[NEWLINE\left|\left\langle(Df(z))^{-1}D^2f(z)(z,w(z)), \frac{\partial h^2}{\partial\overline z}(z)\right\rangle\right|\leq M(z)\left|\left\langle w(z), \frac{\partial h^2}{\partial\overline z}(z) \right\rangle\right|NEWLINE\]NEWLINE for \(z\in\Omega \setminus\{0\}\), where \(w(z) = (Df(z))^{-1}(f(z))\) and \(M(z) < 1\). Then \(f\) is starlike.