Boundary logarithmic mean of a univalent function

If $\varphi$ maps the unit disc conformally to a proper simply connected domain, the nonvanishing holomorphic quotient $F(w)=(\varphi (w)-\varphi (0))/w$ has harmonic log modulus. Its radial mean equals $\log |{\varphi }^{\prime }(0)|$. Boundary values are understood as radial limits: the standard integral-mean bound ${\sup }_{r<1}\int |\varphi (r{e}^{i\theta }){|}^{p}d\theta <\infty$ for $0<p<1/2$, together with the Koebe distortion theorem lower bound $|F(w)|\ge |{\varphi }^{\prime }(0)|/4$, gives uniform integrability of these logarithms. Hence