= Jensen's formula
{c}
{title2=$\log|f(0)|+\sum_{|a_j|<R}\log(R/|a_j|)=\tfrac1{2\pi}\int_0^{2\pi}\log|f(Re^{i\theta})|\,d\theta$}
{wiki}
Let $R>0$ and let $f$ be a <holomorphic function> on an open neighbourhood of the closed <disk> $\{z:|z|\le R\}$. Assume $f(0)\ne0$ and that $f$ has no <zeros of a function> on the boundary <circle> $|z|=R$. List its interior <zeros of a function> as $a_1,\ldots,a_N$, repeating each according to its <multiplicity of a zero>. With the natural <logarithm>, <Jensen's formula> states
$$
\frac{1}{2\pi}\int_0^{2\pi}\log|f(Re^{i\theta})|\,d\theta
=\log|f(0)|+\sum_{j=1}^{N}\log\frac{R}{|a_j|}.
$$
The empty sum is zero. This boundary-zero-free version is the holomorphic special case of the general identity in the source.<246B, Notes 1: Zeroes, poles, and factorisation of meromorphic functions by Terence Tao>[§1, Theorem 2 and equation (6): neighbourhood of the closed disk, zeros counted with multiplicity, holomorphic special case]{ref}
Every interior <zero of a function> makes a positive contribution; a zero nearer the centre contributes more. Consequently boundary growth relative to $|f(0)|$ limits how many <zeros of a function> can lie well inside the <disk>. The <Jensen zero-count bound> makes this quantitative, while the <integrated zero count> retains the individual radial weights. The boundary statistic here is the <circle log-mean of a holomorphic function>; exponentiating gives a <circle geometric mean of a holomorphic function>.
Back to article page
