A problem on the Bloch norm of functions in Doob’s class

Abstract:

Let $\Delta$ denote the unit disc and $\partial \Delta$ denote the unit circle both in the complex plane. Define the Doob’s class $D(\rho )$, $0 < \rho < 2\pi$, as all holomorphic functions on $\Delta$ satisfying (1) $f(0) = 0$, and (2) for some arc ${\Gamma _f} \subseteq \partial \Delta$ with arclength $\rho$, for all $p \in \Gamma$, ${\underline {\lim } _{z \to p}}|f(z)| \geq 1$. Recently the author and Rung [6] proved a conjecture of Doob made in 1935 by showing that the norm \[ ||f|| = {\sup _{z \in \Delta }}(1 - |z{|^2})|f’(z)| \geq \frac {{2\sin \theta (\rho )}}{{e\theta (\rho )}},\quad 0 \leq \theta (\rho ) \leq \pi - \rho /2.\] We then conjecture that the result should be true if the arc ${\Gamma _f}$ is replaced by a finite union of arcs whose total length is at least $\rho$. In this paper, we answer this problem. It turns out to be surprising that the answer depends on the connectivity of the union, namely, the answer is no for the disconnected case, but yes for the connected one.