TITLE: A qualifying exam problem involving Schwarz lemma
QUESTION [5 upvotes]: This is a problem in the book "Berkeley Problems in Mathematics", which I think the solution given is wrong, can someone help?
The following problem appeared in Spring 1991.

Let the function $f$ be analytic in the unit disc, with $|f(z)|\leqslant 1$ and $f(0)=0$. Assume that there is a number $r$ in $(0,1)$ such that $f(r)=f(-r)=0$. Prove that
$$|f(z)|\leqslant |z|\left| \frac{z^2-r^2}{1-r^2z^2}\right|.$$

REPLY [5 votes]: Note that the right hand side of the inequality is equal to $1$ on the unit circle.  Then use the maximum modulus principle on the holomorphic function $$ \frac{f(z)(1-r^2z^2)}{z(z^2-r^2)}.$$