= Integrated zero count
{title2=$N_f(R)=\int_0^R\frac{n_f(t)}{t}\,dt$}
Under the hypotheses of <Jensen's formula>, $f(0)\ne0$ ensures that the <radial zero-counting function> vanishes for sufficiently small positive $t$. Integrating each zero's contribution separately gives
$$
N_f(R)=\int_0^R\frac{n_f(t)}{t}\,dt
=\sum_{|a|<R}\operatorname{ord}_a f\int_{|a|}^{R}\frac{dt}{t}
=\sum_{|a|<R}\operatorname{ord}_a f\log\frac{R}{|a|}
=L_f(R)-\log|f(0)|.
$$
The values of $n_f$ at its finitely many jump points do not change the <integral>. For $0<r<R$, every zero in the smaller <open disc> supplies at least $\log(R/r)$ per unit of <multiplicity of a zero>, so
$$
n_f(r)\log(R/r)\le N_f(R)\le\log M_f(R)-\log|f(0)|,
\qquad M_f(R)=\max_{|z|=R}|f(z)|.
$$
The second inequality bounds the <circle log-mean of a holomorphic function> by the <logarithm> of the largest boundary <complex modulus>, recovering the <Jensen zero-count bound>. An upper bound on boundary growth and a fixed nonzero centre value therefore constrain interior zeros. The factor $\log(R/r)$ also explains why the same estimate becomes weaker as the smaller radius approaches the boundary.
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