Let $\Bbb D= \{z : |z|<1\}$ be the unit disc. Let $f:\Bbb D \to \Bbb D$ be a holomorphic function such that $|f(\frac{1}{2})|+|f(-\frac{1}{2})|=0$. Prove: $|f(0)| \leq \frac{1}{4}$.
The only thing I thought about was using Schwarz lemma, but I'm not sure how. I thought about defining $g(z)=\frac{f(z+\frac{1}{2})+f(z-\frac{1}{2})}{2}$, but now $g$ is not defined in all $\Bbb D$, and I'm pretty much stuck.
In the question's current form (i.e. there are zeroes at $\frac{1}{2}$ and $\frac{-1}{2}$), the claim immediately follows from Jensen's formula (applied with $r$ arbitrarily close to $1$):
$\log |f(0)| = \frac{1}{2\pi} \int_0^{2\pi} \log |f(re^{i\theta})|d\theta + \sum \log(\frac{|a_k|}{r})$ where $r$ is s.t. $f$ is analytic in an open set containing $|z| \le r$, and the sum is over all zeroes of $f$ in the open disk $|z| < r$.