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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

An inequality for the norm of a polynomial factor


Author: Igor E. Pritsker
Journal: Proc. Amer. Math. Soc. 129 (2001), 2283-2291
MSC (1991): Primary 30C10, 30C85; Secondary 11C08, 31A15
Published electronically: November 30, 2000
MathSciNet review: 1823911
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Abstract:

Let $p(z)$ be a monic polynomial of degree $n$, with complex coefficients, and let $q(z)$ be its monic factor. We prove an asymptotically sharp inequality of the form $\Vert q\Vert _{E} \le C^n \, \Vert p\Vert _E$, where $\Vert\cdot\Vert _E$ denotes the sup norm on a compact set $E$ in the plane. The best constant $C_E$ in this inequality is found by potential theoretic methods. We also consider applications of the general result to the cases of a disk and a segment.


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Additional Information

Igor E. Pritsker
Affiliation: Department of Mathematics, 401 Mathematical Sciences, Oklahoma State University, Stillwater, Oklahoma 74078-1058
Email: igor@math.okstate.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05818-4
PII: S 0002-9939(00)05818-4
Keywords: Polynomials, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points
Received by editor(s): November 8, 1999
Received by editor(s) in revised form: November 23, 1999
Published electronically: November 30, 2000
Additional Notes: Research supported in part by the National Science Foundation grants DMS-9996410 and DMS-9707359.
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society