The curvature of level curves

Abstract

Sharp bounds are derived for the curvature of level curves of analytic functions in the complex plane whose logarithmic derivative has the representation c / ( w − g ( w ) ) c/(w - g(w)) , where g ( w ) g(w) is analytic for | w | > a |w| > a and | g ( w ) | ≦ a , c |g(w)| \leqq a,c real. These results are applied in particular to lemniscates and sharpened for the level curves of lacunary polynomials. Extensions to the level curves of Green’s function and rational functions are indicated.