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The Areas of Polynomial Images and Pre‐Images
Author(s) -
Crane Edward
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003509
Subject(s) - mathematics , monic polynomial , multiplicity (mathematics) , mathematics subject classification , isoperimetric inequality , degree (music) , polynomial , image (mathematics) , combinatorics , plane (geometry) , pure mathematics , mathematical analysis , geometry , artificial intelligence , physics , computer science , acoustics
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 π n (Area( K )/π) n , and the area of the pre‐image of K under p is at most π 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. 2000 Mathematics Subject Classification 30C10 (primary), 26D05, 30C85 (secondary).