Estimate the number of roots of an analytic function Let $f : \mathbb{C}\longrightarrow \mathbb{C}$ be analytic with $0 \not = f(0)$. Suppose we have normalized $f$ such that $|f(0)| = 1$. Suppose that $f$ has $n$ roots (including repeated roots) and they are all in $B_{\frac 12}(0)$. Is it possible to estimate $n$ in terms the maximum value of $f$ on the unit circle  !?

By Jensen's formula, $n\leq \frac{\log M}{\log2}$, where $M = \sup\limits_{\vert z\vert =1}\vert f(z)\vert$. 
https://en.wikipedia.org/wiki/Jensen%27s_formula
