SCHWEITZER 1953 Problem 9

Let $w=f(x)$ be regular in $\left | z \right |\leq 1$ . For $0\leq r \leq 1$ , denote by c, the image by $f(z)$ of the circle $\left | z \right | = r$ . Show that if the maximal length of the chords of $c_{1}$ is $1$ , then for every $r$ such that $0\leq r \leq 1$ , the maximal length of the chords of c, is not greater than $r$ . (F. 1)