An improved estimate for the Bloch norm of functions in Doob’s class

Abstract:

For any fixed $0 < \rho < 2\pi ,\mathcal {D}(\rho )$ is the family of all holomorphic functions in $\Delta$ which satisfy (i) $f(0) = 0$, and (ii) ${\underline {\lim } _{z \to \tau }}|f(z)| \geqslant 1$, for all $\tau$ lying on some arc ${\Gamma _f} \subseteq \partial \Delta$ with arclength $|{\Gamma _f}| \geqslant \rho$. We showed that for each $f \in \mathcal {D}(\rho )$ there exists a point ${z_f} \in \Delta$ at which \[ |f’({z_f})|(1 - |{z_f}{|^2}) \geqslant \frac {2}{e}\frac {{\sin (\pi - (\rho /2))}}{{(\pi - (\rho /2))}}.\] In this paper we improve this estimate by replacing the quantity $\pi - (\rho /2)$ with a value $\theta (\rho )$ which lies between 0 and $\pi - (\rho /2)$ and so improves the estimate. The value $\theta (\rho )$ is defined as the (unique) solution in this interval of the equation ${F_\rho }(\theta ) = \log (\cot (\rho /4)\cot (\theta /2)) - \theta /\sin \theta = 0.$.