= Boundary logarithmic mean of a univalent function
If $\phi$ maps the unit disc conformally to a proper simply connected domain, the nonvanishing holomorphic quotient $F(w)=(\phi(w)-\phi(0))/w$ has harmonic log modulus. Its radial mean equals $\log|\phi'(0)|$. Boundary values are understood as radial limits: the standard integral-mean bound $\sup_{r<1}\int|\phi(re^{i\theta})|^p\,d\theta<\infty$ for $0<p<1/2$, together with the <Koebe distortion theorem> lower bound $|F(w)|\geq|\phi'(0)|/4$, gives <uniform integrability> of these logarithms. Hence
$$
\log|\phi'(0)|=\frac1{2\pi}\int_0^{2\pi}\log|\phi(e^{i\theta})-\phi(0)|\,d\theta.
$$
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