Spijker's lemma states that the image of any circle under a complex rational
function whose numerator and denominator
have polynomial degrees at most 
has length at most 
on the Riemann sphere
of radius 1, with length counted with traversal multiplicity.
Here 
is a positive integer, and the zero polynomial
is allowed as the numerator (Wegert and Trefethen 1994,
pp. 132-133).
Spijker's Lemma
Explore with Wolfram|Alpha
References
Edelman, A. and Kostlan, E. "How Many Zeros of a Random Polynomial Are Real?" Bull. Amer. Math. Soc. 32, 1-37, 1995.Wegert, E. and Trefethen, L. N. "From the Buffon Needle Problem to the Kreiss Matrix Theorem." Amer. Math. Monthly 101, 132-139, 1994.Referenced on Wolfram|Alpha
Spijker's LemmaCite this as:
Weisstein, Eric W. "Spijker's Lemma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SpijkersLemma.html