Let $f:\mathbb{D}\to \mathbb{D}$ be a holomorphic function on the unit disc. If $z_1,\cdots,z_n\in D$ are the zeros of $f$, then $$|f(0)|\leq |z_1||z_2|\cdots |z_n|.$$
My attempts
Let $$f(z)=(z-z_1)(z-z_2)\cdots(z-z_n)\sum_{n=0}^{\infty}a_nz^n,$$
then $f(0)=(z_1\cdots z_n)\cdot a_0$. Hence, we need to show that $|a_0|\leq 1.$
How to use $f(\mathbb{D})\subset \mathbb{D}$ to prove this result? And one may notice that $g(z)=\sum a_nz^n$ has no zero in $\mathbb{D}$.
