= Solution
Let $f$ be a nonzero <meromorphic function> in a neighborhood of the closed disk $|z|\le R$, with no boundary zero or <pole>. List its interior zeros $a_j$ with <multiplicities> $m_j$ and its interior <poles> $b_k$ with orders $n_k$. Define the disk <Green function> and <Poisson kernel> by
$$
G_R(z,a)=\log\left|\frac{R^2-\overline a z}{R(z-a)}\right|,\qquad P_R(z,\theta)=\frac{R^2-|z|^2}{|Re^{i\theta}-z|^2}.
$$
At every interior point that is neither a zero nor a <pole>, the <Poisson-Jensen formula> is
$$
\boxed{\log|f(z)|=\frac1{2\pi}\int_0^{2\pi}P_R(z,\theta)\log|f(Re^{i\theta})|\,d\theta-\sum_jm_jG_R(z,a_j)+\sum_kn_kG_R(z,b_k).}
$$
In particular, at $z=0$ when $f(0)$ is finite and nonzero, this becomes <Jensen's formula>, with each zero contributing $-m_j\log(R/|a_j|)$ and each <pole> contributing $n_k\log(R/|b_k|)$.
Here is a proof that also explains the signs. The disk <Blaschke factors>
$$
B_a(z)=\frac{R(z-a)}{R^2-\overline a z}
$$
have a simple zero at $a$, no <poles> in the disk and boundary <modulus> one. Therefore
$$
F(z)=f(z)\frac{\prod_kB_{b_k}(z)^{n_k}}{\prod_jB_{a_j}(z)^{m_j}}
$$
extends to a zero-free <holomorphic function> throughout the disk. Its logarithmic <modulus> is a <harmonic function> and agrees with $\log|f|$ on the boundary. The <Poisson integral> represents $\log|F(z)|$ by that boundary data. Since $\log|B_a|=-G_R(\cdot,a)$, substitution gives the displayed <Poisson-Jensen formula>. Boundary zeros or <poles> can be handled by taking limiting radii: their logarithmic singularities are integrable. When zero is itself a zero or <pole>, factor out its signed power of $z$ before evaluating <Jensen's formula> there.
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