Univalence criteria for holomorphic mappings in \(\mathbb{C}^n\) (Q2711260)

Let \(B^n\) be the open unit Euclidean ball in \(\mathbb{C}^n\), let \(H(B^n)\) be the class of holomorphic mappings from \(B^n\) into \(\mathbb{C}^n\) and let \(I\) be the identity in \({\mathcal L}(\mathbb{C}^n)\).NEWLINENEWLINENEWLINEIn this paper the authors obtain the following sufficient condition of univalence, using the method of subordination chains:NEWLINENEWLINENEWLINETheorem 2. Let \(f,g\in H(B^n)\) such that \(f(0)= g(0)= 0\), \(Df(0)= Dg(0)= I\) and \(g\) is locally biholomorphic in \(B^n\). Let \(a: [0,\infty)\to \mathbb{C}\) be a function satisfying the following conditions:NEWLINENEWLINENEWLINE1) \(a\in C^1[0,\infty)\), \(a(0)= 1\), \(a(t)\neq 0\), \(t\in [0,\infty)\);NEWLINENEWLINENEWLINE2) \(\lim_{t\to\infty}|a(t)|= \infty\);NEWLINENEWLINENEWLINE3) \(\text{Re}{a'(t)\over a(t)}> 0\), \(t\geq 0\).NEWLINENEWLINENEWLINEIf NEWLINE\[NEWLINE\Biggl\|[Dg(z)]^{-1} Df(z)- {1+ a'(0)\over 2} I\Biggr\|< {|1+ a'(0)|\over 2},\quad z\in B^nNEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{multlined} \max_{\|z\|= e^{-t}}\Biggl\|\|z\|[Dg(z)]^{-1} Df(z)+ (a(t)- \|z\|)[Dg(z)]^{-1} D^2 g(z)(z,\cdot)+\\ \Biggl[{a(t)- a'(t)\over 2}-\|z\|\Biggr] I\Biggr\|< {|a(t)+ a'(t)|\over 2}\end{multlined}NEWLINE\]NEWLINE for all \(t\geq 0\), then \(f\) is univalent on \(B^n\).