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The Area of Polynomial Images and Preimages

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arxiv math/0302189 v1 pith:GPNPCEW5 submitted 2003-02-17 math.CV math.CAmath.MG

Edward Crane (Trinity College , Cambridge) This is my paper · ORCID
classification math.CVmath.CAmath.MG
keywords areaboundcomplexinequalityplanepolynomialunderbounds
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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.

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