Length functions of lemniscates
Complex Variables
2009-03-01 v2 Differential Geometry
Abstract
We study metric and analytic properties of generalized lemniscates E_t(f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E_t(f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E_t(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.
Cite
@article{arxiv.math/0306327,
title = {Length functions of lemniscates},
author = {Olga Kuznetsova and Vladimir Tkachev},
journal= {arXiv preprint arXiv:math/0306327},
year = {2009}
}
Comments
18 pages