The Area of Polynomial Images and Preimages
Complex Variables
2007-05-23 v1 Classical Analysis and ODEs
Metric Geometry
Abstract
Let p be a monic polynomial in one complex variable and K a measurable subset of the complex plane. In terms of the area of K, we give an upper bound on the area of the preimage of K under p and a lower bound on the area of the image of K under p, (counted with multiplicity). Both bounds are sharp. The former extends an inequality of Polya. The proof uses Carleman's isoperimetric inequality for plane condensers. We include a summary of the necessary potential theory.
Keywords
Cite
@article{arxiv.math/0302189,
title = {The Area of Polynomial Images and Preimages},
author = {Edward Crane},
journal= {arXiv preprint arXiv:math/0302189},
year = {2007}
}
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8 pages