Supremum over inverse image of functions in the Bloch space (Q875309)

Let \(\mathbb D\) be the unit disk in the complex plane. A function \(f\) is called an \(\alpha\)-Bloch function if it is analytic on \(\mathbb D\) and \(\| f\|_{\alpha} =\sup_{z\in\mathbb D}(1-| z| ^2)^{\alpha}| f'(z)| <\infty\). As a positive result in the paper under review the author proves the following result: For \(\alpha>-1, \exists\) a constant \(K(\alpha)>0\) depending only on \(\alpha\) such that \[ \sup_{z\in f^{-1}(\Sigma_{\epsilon})}(1-| z| ^2)^{\alpha}| f'(z)| \geq K(\alpha)\epsilon\frac{| f'(0)| ^4}{\| f\|_{\alpha}^3} \] for any nonconstant \(f\in\mathcal B^{\alpha}\) with \(f(0)=0\), where \(\Sigma_{\epsilon} = \{w\in\mathbb C:\,| \arg w| <\epsilon\}\). However, for \(\alpha=-1\), it is shown that this kind kind of conclusion is not valid any more.