= Polynomial circle log-mean
{title2=$L_p(R)=\log|c|+\sum_j\log\max(R,|a_j|)$}
For a <polynomial> $p(z)=c\prod_j(z-a_j)$ with $c\ne0$, no <polynomial root> at zero, and $R>0$ different from every $|a_j|$, <Jensen's formula> gives the displayed expression. The <polynomial roots> are repeated according to their <multiplicity of a zero>: an exterior root contributes $\log|a_j|$, whereas an interior root contributes $\log R$.
For an explicit example, take the <polynomial>
$$
f(z)=3(z-\tfrac14)(z+\tfrac12),\qquad f(0)=-\tfrac38.
$$
Its two <simple zeros> are $1/4$ and $-1/2$. The <circle log-mean of a holomorphic function> is therefore
$$
L_f(R)=\begin{cases}
\log(3/8),&0<R<1/4,\\
\log(3R/2),&1/4<R<1/2,\\
\log(3R^2),&R>1/2.
\end{cases}
$$
At $R=0.2$ the <open disc> contains no <zeros of a function>, at $R=0.4$ it contains $1/4$ only, and at $R=0.75$ it contains both. These three <circles> avoid the <polynomial roots>; the radii $1/4$ and $1/2$ are excluded from the displayed version.
\Image[/jensen-polynomial-zero-disks.png]
{title=The zeros of $3(z-1/4)(z+1/2)$ inside three concentric disks}
{description=The three boundary <circles> enclose respectively zero, one and two <simple zeros>. Each zero contributes to <Jensen's formula> only after the radius exceeds its <complex modulus>.}
The corresponding values of the <circle log-mean of a holomorphic function> and the <circle geometric mean of a holomorphic function> are
$$
\begin{array}{c|c|c|c}
R&n_f(R)&L_f(R)&G_f(R)\\\hline
0.2&0&\log(3/8)\approx-0.980829&3/8\\
0.4&1&\log(3/5)\approx-0.510826&3/5\\
0.75&2&\log(27/16)\approx0.523248&27/16
\end{array}
$$
For example, at $R=0.4$ the contribution of the sole interior <zero of a function> is $\log(0.4/0.25)=\log(8/5)$; adding it to $\log(3/8)$ gives $\log(3/5)$. The second <polynomial root> still affects $f(0)$, but contributes no additional interior-zero term at this radius.
\Image[/jensen-polynomial-circle-log-mean.png]
{title=Circle log-mean versus log-radius for $3(z-1/4)(z+1/2)$}
{description=The <log-radius slope of the circle log-mean> is zero, one and two in the three intervals, matching the <radial zero-counting function>. Open markers exclude the boundary-zero radii $1/4$ and $1/2$; filled markers show the three computed examples.}
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