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On an Inequality of S. Bernstein

Published online by Cambridge University Press:  20 November 2018

C. Frappier  and
Q. I. Rahman
C. Frappier
Affiliation:
Université de Montréal, Montréal, Québec
Q. I. Rahman
Affiliation:
Université de Montréal, Montréal, Québec
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Let R > 1 and denote by ℰR the ellipse

(1)

If Pn is a polynomial of degree at most n such that

(2)

then ([2]; also see [13, p. 337] and [9, p. 158, Prob. No. 270])

(3)

The standard proof of this well known result runs as follows. The function

(4)

is entire and in view of (2) we have

Hence by the maximum modulus principle

(5)

which is clearly equivalent to (3).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 932 - 944
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

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