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On Bernstein's Inequality

Published online by Cambridge University Press:  20 November 2018

A. Giroux ,
Q. I. Rahman  and
G. Schmeisser
A. Giroux
Affiliation:
Université de Montréal, Montréal, Québec
Q. I. Rahman
Affiliation:
Universitàt Erlangen Nünberg, Erlangen, West Germany
G. Schmeisser
Affiliation:
Universitàt Erlangen Nünberg, Erlangen, West Germany
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1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])

(1)

Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdôs and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by

(2)

On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then

(3)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 31 , Issue 2 , 01 April 1979 , pp. 347 - 353
Copyright
Copyright © Canadian Mathematical Society 1979

References

1. Giroux, A. and Rahman, Q. I., Inequalities for polynomials with a prescribed zero, Trans. Amer. Math. Soc. 193 (1974), 67–98.Google Scholar
2. Lax, P. D., Proof of a conjecture of P. Erdôs on the derivative of a polynomial, Bull. Amer. Math. Soc. 50 (1944), 509–513.Google Scholar
3. Malik, M. A., On the derivative of a polynomial, J. London Math. Soc. 1 (1969), 57–60.Google Scholar
4. Schaeffer, A. C., Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565–579.Google Scholar
5. Turân, P., Uber die ableitung von polynomen, Compositio Math. 7 (1939-40), 89–95.Google Scholar