$f$ holomorphic from unit disc to itself. $f(1/2) = f(-1/2) = 0$. Show that $|f(0)| \le 1/3$.

$f$ holomorphic from unit disc to itself. $f(1/2) = f(-1/2) = 0$. Show
that $|f(0)| \le 1/3$.

I'm studying for a qual exam. I cannot solve the following problem:
Let $f$ be holomorphic from the unit disc to itself. $f(1/2) = f(-1/2) =
0$. Show that $|f(0)| \le 1/3$.
With an application of Schwarz lemma, I showed that $|f(0)| \le 1/2$. I
can't prove the tighter bound. Any ideas?