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The Area of Polynomial Images and Preimages

Edward Crane
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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.

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Edward Crane. 2003-02-17. The Area of Polynomial Images and Preimages. https://arxiv.org/abs/math/0302189

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