Anderson's Conjecture and Radial Variation of Bloch Functions

In 1971 J.M. Anderson conjectured that for any conformal map $\phi$ in the unit disc there exists $\beta ,0\le \beta \le 2\pi$ such that $${\int }_0^1|\phi ‘‘(r{e}^{i\beta })|dr<\mathrm{\infty }$$ More recently this problem has been posed in the works of N. Makarov, Ch. Pommerenke and D. Gnuschke - Ch. Pommerenke. The purpose of the present talk is to prove Anderson's conjecture. This will be done by showing the following theorem about the associated Bloch function $b=\log |\phi ‘|$.
Theorem 1 There exists $\beta$,$0\le \beta \le 2\pi$ such that $$b\left(r{e}^{i\beta }\right)\le -\delta {\int }_0^r|\mathrm{\nabla }b\left(\rho e^{i\beta }\right)|d\rho +\frac{1}{\delta }\$,for\$0<r<1$$ where $\delta >0$ is independent of $r<1$.