Jensen's Formula and the number of zeros. I am doing a question that requires me to find the number of zeros in a disk when $f$ is non-constant, bounded and analytic on $D$. I saw that the Wikipedia page of Jensen's formula has the following information:
Jensen's formula can be used to estimate the number of zeros of analytic function in a circle. Namely, if f is a function analytic in a disk of radius R centered at $z_0$ and if |f| is bounded by M on the boundary of that disk, then the number of zeros of f in a circle of radius r < R centered at the same point $z_0$ does not exceed

Does anyone know how this formula is derived?

Let $f(z)$ be analytic function such that $f(0)\ne0$, and let $z_1,z_2,\dots,z_m$ denote the zeros of $f(z)$ satisfying $|z_k|<R$. Then Jensen's formula gives
$$
\int_0^1\log|f(Re^{2\pi it})|\mathrm dt=\log\left|f(0)\cdot{R\over z_1}\cdot{R\over z_2}\cdots{R\over z_m}\right|
$$
Now, let $|f(z)|\le M$ for $|z|\le R$, then we have
$$
\log\left|{R\over z_1}\cdot{R\over z_2}\cdots{R\over z_m}\right|\le\log M-\log|f(0)|
$$
Let $0<r<R$, so we can classify $z_1,z_2,\dots,z_m$ into two classes:
$$
\log\left|{R\over z_1}\cdot{R\over z_2}\cdots{R\over z_m}\right|
=\sum_{|z_k|\le r}\log\left|R\over z_k\right|+\sum_{|z_j|>r}\log\left|R\over z_j\right|
$$
Because $\log|R/z_k|$ is at least $\log R/r$, we have
$$
N(r)\log\frac Rr\le\sum_{|z_k|\le r}\log\left|R\over z_k\right|
$$
where $N(r)$ denotes the number of zeros of $f(z)$ satisfying $|z|\le r$. Now, by the transitivity of inequalities, we conclude that the number of zeros of $f(z)$ within $|z|\le r$ satisfies the following inequality:
$$
N(r)\le{1\over\log R/r}\log\left|M\over f(0)\right|
$$
Hope this will help you!
