Loewner chains and univalence criteria (Q2711261)

Let \(B^n\) be the open unit Euclidean ball in \(\mathbb{C}^n\) and let \({\mathcal H}(B^n)\) be the class of holomorphic mappings from \(B^n\) into \(\mathbb{C}^n\). Also let \(I_n\) be the identity in \({\mathcal L}(\mathbb{C}^n)\).NEWLINENEWLINENEWLINEThe main result of this paper is Theorem 2, obtained by using the method of subordination chains:NEWLINENEWLINENEWLINELet \(f\in{\mathcal H}(B^n)\), \(f(0)= 0\), \(Df(0)= I_n\), be locally univalent in \(B^n\), let \(c\in \mathbb{C}\setminus\{-1\}\) with \(|c|\leq 1\) and let \(\alpha\) be a real number, \(\alpha\geq 2\). If NEWLINE\[NEWLINE\Biggl\|c\|z\|^\alpha I_n+ (1-\|z\|^\alpha) [Df(z)]^{-1} D^2f(z)(z,\cdot)- \Biggl({\alpha\over 2}- 1\Biggr) I_n\Biggr\|< {\alpha\over 2},\quad z\in B^n,NEWLINE\]NEWLINE then \(f\) is univalent on \(B^n\).