Jensen's Inequality (Complex Analysis)

Theorem
Let $R>0$ be a real number.

Let $f$ be a complex valued function that is analytic in the disk of radius $R$ centered at $0$.

Suppose $|f(z)|\leq M$ for $|z|\leq R$.

Suppose $f(0)\neq0$.

Let $0<r<R$.

Then the number of zeroes of $f$ in the disk of radius $r$ centered at $0$ is at most
 * $\dfrac{\log(M/|f(0)|)}{\log(R/r)}$.