I have just encountered the following exercise which has me quite stumped:
Let $ f $ be analytic on the closed unit disk, and we assume that $ | f(z) | \leq 1 $ for all $ z $'s in this set. We also have $ f(\frac{1}{2}) = f(\frac{i}{2}) = 0 $. We need to prove that $ |f(0)| \leq \frac{1}{4} $.
I thought about using Schwarz's lemma to modify the function, but in Schwarz's lemma we assume only one zero at the origin, how does one use the two zeroes here to modify the function and obtain the bound?
Any help would be appreciated.