Length functions of lemniscates
Olga Kuznetsova, Vladimir Tkachev
Abstract
We study metric and analytic properties of generalized lemniscates Et(f)=z:ln|f(z)|=t, where f is an analytic function. Our main result states that the length function |Et(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|Et(f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulas of the length function in some special cases.
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