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THE AREAS OF POLYNOMIAL IMAGES AND PRE-IMAGES

Published online by Cambridge University Press:  19 October 2004

EDWARD CRANE
Affiliation:
Trinity College, Cambridge CB2 1TQ, United Kingdom Current address: Merton College, Oxford OX1 4JD, United [email protected]
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Abstract

Let $p$ be a monic complex polynomial of degree $n$, and let $K$ be a measurable subset of the complex plane. Then the area of $p(K)$, counted with multiplicity, is at least $\pi n ( \Area (K)/\pi )^n$, and the area of the pre-image of $K$ under $p$ is at most $\pi^{1-1/n} (\Area (K) )^{1/n}$. Both bounds are sharp. The special case of the pre-image result in which $K$ is a disc is a classical result, due to Pólya. The proof is based on Carleman's classical isoperimetric inequality for plane condensers.

Type
Papers
Copyright
© The London Mathematical Society 2004

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