Nevanlinna proximity function ($m(r,f)=\frac{1}{2\pi }{\int }_0^{2\pi }{\log }^{+}|f(r{e}^{i\theta })|d\theta$)

Here ${\log }^{+}x=\max (0,\log x)$. The function $m(r,f)$ measures logarithmic proximity of $f$ to infinity. For a finite target $a$, proximity is $m(r,1/(f-a))$. Logarithmic singularities on a circle are integrable.