A distortion theorem and the Bloch constant for Bloch mappings in \(\mathbb{C}^N\)
Denote by \(B_X\) the unit ball in a complex Banach space \(X\). Let \[ c(B_X)=\frac{1}{2} \sup_{ x,y\in B_X} |h_0(x, y)|, \] where \(h_0\) is the Bergman metric on \(B_X\) at \(0\). \par Theorem 1. Let \(B_X\) be the unit ball of an \(n\)-dimensional JB\(^*\)-triple \(X.\) Let \(\alpha \in (0, 1]\) and let \(m(\alpha)\) be the unique root of the equation \[ t(1 - t^2)^{c(B_X)} \Big(\frac{2c(B_X)+1}{2c(B_X)}\Big)^{c(B_X)}\sqrt{2c(B_X)+1}=\alpha \] in the interval \([0, \frac{1}{\sqrt{2c(B_X)+1}}].\) If \(f\in H(B_X, C^n),\) \(\|f \|_0 =1\) and \(\det Df(0) = \alpha\), then \[ |\det Df(z)|\geq \Re \det Df(z) \geq \frac{a}{m(\alpha)} \frac{m(\alpha)-\|z\|}{(1 - m(\alpha)\|z\|)^{2c(B_X)+1}} \] for \[ \|z\|\leq \frac{1+\sqrt{2c(B_X)+1}m(\alpha)}{\sqrt{2c(B_X)+1}+m(\alpha)}, \]and \[ \det Df(z) \geq \frac{\alpha}{m(\alpha)} \frac{m(\alpha)-\|z\|}{(1 - m(\alpha)\|z\|)^{2c(B_X)+1}} \] for \[ \|z\|\leq \frac{1-\sqrt{2c(B_X)+1}m(\alpha)}{\sqrt{2c(B_X)+1}-m(\alpha)}. \] Moreover, the above estimates are sharp. Let \(B_X\) be the unit ball of an \(n\)-dimensional \(\mathrm{JB}^*\)-triple \(X\). \(f\in H(B_X,\mathbb C^n)\) is called a Bloch mapping if the family \[ F_f=\big\{g(z)=f(\varphi(z))-f(\varphi(0)):\varphi\in\mathrm{Aut}(B_X)\big\} \]is a normal family. The Bloch semi-norm of a Bloch mapping \(f\) is defined as \[ \Vert f\Vert_B=\sup\big\{\Vert D(f\circ\varphi)(0)\Vert_X: \varphi\in\mathrm{Aut}(B_X)\big\}. \]Let \(\beta(K)\) denote the set of Bloch mappings \(f\) with \(\Vert f\Vert_B=K\), where \(1\le K\le\infty\). Let \(B^n\) be the Euclidean unit ball of \(\mathbb C^n\). For \(f\in H(B_X,\mathbb C^n)\), a schlicht ball of \(f\) centered at \(f(a)\) is a Euclidean ball with center \(f(a)\) such that \(f\) maps an open subset of \(B_X\) containing \(a\) biholomorphically onto this ball. For a point \(a\in B_X\), let \(r(a,f)\) be the radius of the largest schlicht ball of \(f\) centered at \(f(a)\). \par Theorem 2. Let \(B_X\) be the unit ball of an \(n\)-dimensional JB\(^*\)-triple \(X\). Assume that \(B_X\supset B^n.\) If \(f \in \beta(K),\) \(\|f\|_0 =1\) and \(\det Df(0) = \alpha \in(0,1],\) then \[ r(0, f ) = K^{1-n}\frac{\alpha}{m(\alpha)}\int_0^{m(\alpha)} \frac{(1-t^2)^{n-1}(m(\alpha)-t)}{(1-m(\alpha)t)^{2c(B_X)+1}}dt, \] where \(m(\alpha)\) is the unique root of the equation \[ t(1 - t^2)^{c(B_X)} \Big(\frac{2c(B_X)+1}{2c(B_X)}\Big)^{c(B_X)}\sqrt{2c(B_X)+1}=\alpha \] in the interval \([0, \frac{1}{\sqrt{2c(B_X)+1}}]\).
scientific article; zbMATH DE number 7074801
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