For each positive real number $R$ write $B_{R}=\{z \in \mathbb{C}:|z| \leqslant R\}$. If $F$ is holomorphic on some open set containing $B_{R}$, we define

$$J(F, R)=\frac{1}{2 \pi} \int_{0}^{2 \pi} \log \left|F\left(R e^{i \theta}\right)\right| d \theta$$

1. If $F_{1}, F_{2}$ are both holomorphic on some open set containing $B_{R}$, show that $J\left(F_{1} F_{2}, R\right)=$ $J\left(F_{1}, R\right)+J\left(F_{2}, R\right) .$

2. Suppose that $F(0)=1$ and that $F$ does not vanish on some open set containing $B_{R}$. By showing that there is a holomorphic branch of logarithm of $F$ and then considering $z^{-1} \log F(z)$, prove that $J(F, R)=0$.

3. Suppose that $|w|<R$. Prove that the function $\psi_{W, R}(z)=R(z-w) /\left(R^{2}-\bar{w} z\right)$ has modulus 1 on $|z|=R$ and hence that it satisfies $J\left(\psi_{W, R}, R\right)=0$.

Suppose now that $F: \mathbb{C} \rightarrow \mathbb{C}$ is holomorphic and not identically zero, and let $R$ be such that no zeros of $F$ satisfy $|z|=R$. Briefly explain why there are only finitely many zeros of $F$ in $B_{R}$ and, assuming these are listed with the correct multiplicity, derive a formula for $J(F, R)$ in terms of the zeros, $R$, and $F(0)$.

Suppose that $F$ has a zero at every lattice point (point with integer coordinates) except for $(0,0)$. Show that there is a constant $c>0$ such that $\left|F\left(z_{i}\right)\right|>e^{c\left|z_{i}\right|^{2}}$ for a sequence $z_{1}, z_{2}, \ldots$ of complex numbers tending to infinity.