Let f(t) be a smooth and periodic function of one real variable. Then the
planar curves t↦(f′(t),f(t)) and t↦(f′′(t)−f(t),f′(t)) 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].Comment: 10 pages, 6 figure