A necessary and sufficient condition: Biholomorphic mappings are starlike on a class of Reinhardt domains (Q1179951)

A necessary and sufficient condition for local biholomorphic mappings on Reinhardt domains \[ B\equiv\{z=(z_ 1,z_ 2,\ldots,z_ n)\in\mathbb{C}^ n\mid\;\sum^ n_{i=1}| z_ i|^{p_ i}<1,\;2p_ n>p_ 1\geq p_ 2\geq\cdots\geq p_ n>1\} \] is proved here. The result obtained is: Suppose \(f:B\to\mathbb{C}^ n\) is a holomorphic immersion mapping with \(f(0)=0\). Then \(f\) is starlike if and only if \[ \langle du\circ f^{-1},d\rho\rangle\mid_{W=f(z)}\geq 0 \] holds for any \(Z\in B\) where \(\langle , \rangle\) is the inner product in \(\mathbb{C}^ n\), \(u(Z)=\sum^ n_{i=1}| z_ i|^{p_ i}\) and \(\rho(W)\) is the distance function from the origin in \(\mathbb{C}^ n\).