Let $D$ be the unit disc centered at the origin and f olomorphic in the disc. Show that
$2|f'(0)|\leq sup_ {z,w\in D}|f(z)-f(w)|$
Furthermore, we have equality if and only if f is linear.
I only know Cauchy's estimation for derivatives:
$f^{(n)}(0)\leq \frac{n!}{r^n}Max_{|z|=r} |f|$ for $0 but i've no idea how to apply this to this case.