Spijker's Lemma

The image on the Riemann Sphere of any Circle under a Complex rational mapping with Numerator and Denominator having degrees no more than $n$ has length no longer than $2n\pi$.

References

Edelman, A. and Kostlan, E. ``How Many Zeros of a Random Polynomial are Real?'' Bull. Amer. Math. Soc. 32, 1-37, 1995.