Loewner's "forgotten" theorem

TL;DR

This paper explores Loewner's classical theorem on the non-negative rotation number of certain planar curves derived from smooth periodic functions, providing elementary proofs and an expository overview of the theorem and its historical context.

Contribution

It offers an elementary proof of Loewner's theorem and clarifies its significance through an accessible exposition of the original work.

Findings

Proved that specific planar curves have non-negative rotation number.

Presented an elementary proof following Bol's approach.

Provided an expository overview of Loewner's theorem and its historical background.

Abstract

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].