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Bulletin of the London Mathematical Society (1)

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THE AREAS OF POLYNOMIAL IMAGES AND PRE-IMAGES
EDWARD CRANE
Journal:

Bulletin of the London Mathematical Society / Volume 36 / Issue 6 / November 2004

Published online by Cambridge University Press:

19 October 2004, pp. 786-792

Print publication:

November 2004
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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.