Assume that $f$ is an analytic function on the unit disc $\mathbb{D}$ and continuous up to the closure. Therefore $f(z)=\sum\limits_{n=0}^\infty c_nz^n$ for all $z \in \mathbb{D}$. If $f$ have $m$ zeros in $\mathbb{D}$ how can you prove that $$ \min\{|f(z)| : |z|=1\}\leq |c_0|+\ldots+|c_m| $$
To begin with, the minimum of the funcion in $|z|=1$ shold be equal to the minimun on the whole $\overline{\mathbb{D}}$. Then I tried using the mean value principle around the zeros, or the expansion near a zero, but I couldn't do it.