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On a Problem of Turán about Polynomials II

Published online by Cambridge University Press:  20 November 2018

R. Pierre  and
Q. I. Rahman
R. Pierre
Affiliation:
Université Laval, Québec, Québec
Q. I. Rahman
Affiliation:
Université de Montréal, Montréal, Québec
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1. It was proved by A. A. Markov [3] that if is a polynomial of degree at most n and |pn(x)| ≦ 1 in the interval –1 ≦ x ≦ 1, then in the same interval

(1)

The problem was proposed by the chemist Mendeleev who knew the answer for polynomials of degree 2. For a historical background of the problem see [1].

A. A. Markov's younger brother W. A. Markov considered the problem of determining exact bounds for the j–th derivative of pn(x) at a given point X0 in [ – 1, 1]. His results appeared in a Russian journal in the year 1892; a German version of his remarkable paper was later published in [4]. Amongst other things he proved the following two theorems.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 33 , Issue 3 , 01 June 1981 , pp. 701 - 733
Copyright
Copyright © Canadian Mathematical Society 1981

References

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