  Let $f(t)$ be a smooth and periodic function of one real variable. Then the
planar curves $t\mapsto \big(f'(t),f(t)\big)$ and $t\mapsto
\big(f''(t)-f(t),f'(t)\big)$ both have non-negative rotation number around
every point not on the curve. These are the two simplest cases of a beautiful
Theorem by C. Loewner. This article is expository, we prove the two statements
by elementary means following work by Bol [3]. After that, we present Loewner's
Theorem and his proof from [7].
