On the precise growth, covering, and distortion theorems for normalized biholomorphic mappings (Q596734)

Let \(X\) be a complex Banach space and \(B\) be the unit ball in \(X\). Let \(S(B)\) be the set of biholomorphic mappings \(f\) from \(B\) into \(X\) which are normalized so that \(f(0)=0\) and \(Df(0)=I\), where \(I\) is the identity operator of \(X\) onto \(X\). In this paper the authors obtain sharp growth and covering results, as well as distortion theorems, for mappings \(f\) in various subsets of \(S(B)\) such that \(x=0\) is a zero of order \(k+1\) of \(f(x)-x\). A holomorphic mapping \(f:B\to X\) is called starlike if \(f\) is biholomorphic on \(B\) and \(f(B)\) is a starlike domain with respect to the origin. Let \(S^*(B)\) be the set of normalized starlike mappings on \(B\). One of the main results of this paper is given in theorem 1: If \(f\in S^*(B)\), and \(x=0\) is the zero of order \(k+1\) (\(k\in \mathbb{N}\)) of \(f(x)-x\), then \[ \frac{\| x\| }{(1+\| x\| ^k)^{2/k}}\leq \| f(x)\| \leq\frac{\| x\| } {(1-\| x\| ^k)^{2/k}},\qquad x\in B,\quad f(B)\supset \frac{1}{2^{2/k}}B. \]