Small values of polynomials: Cartan, Pólya and others
<p/> <p>Let <inline-formula><graphic file="1029-242X-1997-104605-i1.gif"/></inline-formula> be a monic polynomial of degree <inline-formula><graphic file="1029-242X-1997-104605-i2.gif"/></inline-formula>, and <inline-formula>&...
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
1997-01-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/1/104605 |
| Summary: | <p/> <p>Let <inline-formula><graphic file="1029-242X-1997-104605-i1.gif"/></inline-formula> be a monic polynomial of degree <inline-formula><graphic file="1029-242X-1997-104605-i2.gif"/></inline-formula>, and <inline-formula><graphic file="1029-242X-1997-104605-i3.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-1997-104605-i4.gif"/></inline-formula>. A classic lemma of Cartan asserts that the lemniscate <inline-formula><graphic file="1029-242X-1997-104605-i5.gif"/></inline-formula> can be covered by balls <inline-formula><graphic file="1029-242X-1997-104605-i6.gif"/></inline-formula>, whose diameters <inline-formula><graphic file="1029-242X-1997-104605-i7.gif"/></inline-formula> satisfy <inline-formula><graphic file="1029-242X-1997-104605-i8.gif"/></inline-formula> For <inline-formula><graphic file="1029-242X-1997-104605-i9.gif"/></inline-formula>, this shows that <inline-formula><graphic file="1029-242X-1997-104605-i10.gif"/></inline-formula> has an area at most <inline-formula><graphic file="1029-242X-1997-104605-i11.gif"/></inline-formula>. Pólya showed in this case that the sharp estimate is <inline-formula><graphic file="1029-242X-1997-104605-i12.gif"/></inline-formula>. We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when <inline-formula><graphic file="1029-242X-1997-104605-i13.gif"/></inline-formula> is normalized to have <inline-formula><graphic file="1029-242X-1997-104605-i14.gif"/></inline-formula> norm 1 on some circle, and Remez' inequality.</p> |
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| ISSN: | 1025-5834 1029-242X |