= Log-radius slope of the circle log-mean
{title2=$\frac{d}{d\log R}L_f(R)=n_f(R)$}
Fix a <holomorphic function> on a neighbourhood of a closed <disk>, with $f(0)\ne0$. On an interval of positive radii whose <circles> contain no <zeros of a function>, differentiating <Jensen's formula> with respect to $x=\log R$ gives
$$
\frac{d}{dx}L_f(e^x)=n_f(e^x).
$$
Each interior <zero of a function> contributes slope one for each unit of its <multiplicity of a zero>. Between consecutive zero moduli, $L_f(e^x)$ is an <affine function>; passing the modulus of a zero of <multiplicity of a zero> $m$ increases the adjacent slopes by $m$. The finite sum $\log|f(0)|+\sum_j\max(0,x-\log|a_j|)$ defines a continuous <convex function> of $x$ whose values agree with the <circle log-mean of a holomorphic function> at every boundary-zero-free radius. This describes the adjacent line segments without asserting a boundary-singularity extension of the <integral>.
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