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For a nonzero holomorphic function, isolated boundary zeros produce only integrable logarithmic singularities: locally a zero of multiplicity $m$ contributes $m\log |\theta -{\theta }_0|$ plus a bounded term. Thus the circle means below are well-defined. The identically zero function may be assigned $J=-\infty$.

1. Multiplication. The pointwise identity $\log |F_1{F}_2|=\log |F_1|+\log |F_2|$ gives $J(F_1{F}_2,R)=J(F_1,R)+J(F_2,R)$ by integration, with the same extended interpretation if a factor is identically zero.

2. Zero-free normalized functions. Compactness lets us choose a slightly larger open disc on which $F$ is holomorphic and zero-free. On this simply connected disc, $F^{\prime }/F$ has a holomorphic primitive $L$ normalized by $L(0)=0$. Differentiating $F{e}^{-L}$ shows it is constant and equal to $F(0)=1$, so $F=e^L$ and $L$ is a holomorphic logarithm of $F$. The Cauchy integral formula givesTaking real parts, using $\Re L=\log |F|$, proves $J(F,R)=0$. For a zero-free function with arbitrary nonzero value at zero, applying this to $F/F(0)$ gives $J(F,R)=\log |F(0)|$.

3. Disc factors. For $|z|=R$,as follows by expansion and $|z{|}^2=R^2$. The denominator's possible zero is at $R^2/\overline{w}$, outside the closed disc; when $w=0$ it is constant. Thus the given Blaschke factor is holomorphic on a neighbourhood of the disc, has modulus one on its boundary and satisfies $J({\psi }_{w,R},R)=0$.

For the general formula, a nonzero entire function has finitely many zeros in the compact disc: an infinite set would have an accumulation point in a holomorphic neighbourhood and the identity theorem would make the function zero identically. List its interior zeros with multiplicities. When $F(0)\ne 0$, divide by their factors ${\psi }_{w_j,R}$ to obtain a holomorphic function $G$ zero-free on a neighbourhood of the disc. The preceding parts give $J(F,R)=J(G,R)=\log |G(0)|$, while ${\psi }_{w_j,R}(0)=-w_j/R$. Hence Jensen's formula isThe source does not explicitly rule out a zero at zero in this general step. If its order there is $m$ and $F(z)=z^m{F}_{\ast }(z)$ with $F_{\ast }(0)\ne 0$, the correct version isThus a formula using $\log |F(0)|$ alone requires the indicated nonzero-centre assumption.

Finally, the lattice hypothesis forces quadratic exponential growth. For $R\ge 8$, all integer pairs with $|j|,|k|\le \lfloor R/4\rfloor$ lie inside the disc of radius $R/2$. Excluding zero, there are at least $R^2/16$ such points, and each contributes at least $\log 2$ to Jensen's sum. Thereforefor boundary-zero-free radii, with the additional nonnegative $m\log R$ term if needed. For all sufficiently large such radii, this is at least $(\log 2)R^2/32$. Choose $R_j\to \infty$ avoiding zero moduli and choose $z_j$ on each circle where $|F|$ is maximal. The logarithm of this circle maximum is at least the logarithmic circle mean, soThis proves lattice zeros force quadratic exponential growth; the smaller constant makes the required inequality strict.