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The Area of Polynomial Images and Preimages | | Edward Crane
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17 Feb 2003 | | Journal: | Bull. London Math. Soc 36 (2004), 786-792. | | Subject: | Complex Variables; Classical Analysis and ODEs; Metric Geometry MSC-class: 30C10 (Primary) 26D05, 30C85 (Secondary) | math.CV math.CA math.MG | | Affiliation: | Trinity College, Cambridge | | 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. | | Source: | arXiv, math.CV/0302189 | | Services: | Forum | Review | PDF | Favorites | |
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