Loewner’s “Forgotten” Theorem

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