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Sharp extensions of Bernstein's inequality to rational spaces

Part of: General properties

Published online by Cambridge University Press:  26 February 2010

Peter Borwein  and
Tamás Erdélyi
Peter Borwein
Affiliation:
Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada, V5A 1S6
Tamás Erdélyi
Affiliation:
Department of Mathematics, Texas A. & M. University, College Station, Texas 77843-3357, U.S.A.
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Abstract

Sharp extensions of some classical polynomial inequalities of Bernstein are established for rational function spaces on the unit circle, on K = r (mod 2 π), on [-1, 1 ] and on ℝ. The key result is the establishment of the inequality

for every rational function f=pn/qn, where pn is a polynomial of degree at most n with complex coefficients and

with | aj | ≠ 1 for each j and for every zo∈ δ D, where δ D,= {z∈ ℂ: |z| = l}. The above inequality is sharp at every z0∈δD.

MSC classification

Secondary: 30A10: Inequalities in the complex domain

Information

Type
Research Article
Information
Mathematika , Volume 43 , Issue 2 , December 1996 , pp. 413 - 423
Copyright
Copyright © University College London 1996

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