Let $f: D(0,1) \to D(0,1) $ a holomorphic map which is also continuous on the boundary of the disk. Let there exists $a\in D(0,1)$ such that $f(a)=f(-a)=0.$
Show that $$|f(0)|\le |a|^2.$$
I don't understand how to use continuous at boundary thing.
Whatever I have done so far is the following : $f$ analytic in and on disk of radius $|a|$ center at $0 . $ Power series of $f(z)$ about $0$ is $$f(z)=a_0+a_1z^2+....$$ $$f(a)=f(-a) \implies f(z)=a_0+a_2z^2+... $$ Now, $f(a)=0 \implies a_0=-(a_2a^2+....)$ So, $|f(0)|=|a^2||(a_2+....)|$ Unable to conclude anything from this . Need help to solve this problem.
