Geodesic
On Bernstein's Inequality
Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 347-353

Voir la notice de l'article provenant de la source Cambridge University Press

1. Introduction and statement of results. If p n(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 p n(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of p n(z). Thus, for example, it was conjectured by P. Erdôs and later proved by Lax [2] that if p n(z) does not vanish in │z│ < 1, then (1) can be replaced by (2) On the other hand, Turán [5] showed that if p n(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then (3) 
Giroux, A.; Rahman, Q. I.; Schmeisser, G. On Bernstein's Inequality. Canadian journal of mathematics, Tome 31 (1979) no. 2, pp. 347-353. doi: 10.4153/CJM-1979-039-3
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[1] 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] 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] 3. Malik, M. A., On the derivative of a polynomial, J. London Math. Soc. 1 (1969), 57–60. Google Scholar

[4] 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] 5. Turân, P., Uber die ableitung von polynomen, Compositio Math. 7 (1939-40), 89–95. Google Scholar

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