# Lower bound in the Bernstein inequality for the first derivative of algebraic polynomials

Brief Communications

First Online:

- 37 Downloads

We prove that max |*p*′(*x*)|, where *p* runs over the set of all algebraic polynomials of degree not higher than *n* ≥ 3 bounded in modulus by 1 on [−1, 1], is not lower than \( {{\left( {n - 1} \right)} \mathord{\left/{\vphantom {{\left( {n - 1} \right)} {\sqrt {1 - {x^2}} }}} \right.} {\sqrt {1 - {x^2}} }} \) for all *x* ∈ (−1, 1) such that \( \left| x \right| \in \bigcup\nolimits_{k = 0}^{\left[ {{n \mathord{\left/{\vphantom {n 2}} \right.} 2}} \right]} {\left[ {\cos \frac{{2k + 1}}{{2\left( {n - 1} \right)}}\pi, \cos \frac{{2k + 1}}{{2n}}\pi } \right]} \).

## Keywords

Approximation Theory Extreme Problem Classical Analysis Chebyshev Polynomial Integral Functional
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

- 1.V. M. Tikhomirov,
*Some Problems of Approximation Theory*[in Russian], Moscow University, Moscow (1976).Google Scholar - 2.B. D. Bojanov, “Markov-type inequalities for polynomials and splines,” in: C. K. Chui, L. L. Schumaker, J. Stökler (editors),
*Approximation Theory, Vol X: Abstract and Classical Analysis*, Vanderbilt University Press (2002), pp. 31–90.Google Scholar - 3.V. A. Markov,
*On Functions Least Deviating from Zero in a Given Interval*[in Russian], St. Petersburg (1892).Google Scholar - 4.L. S. Avvakumova, “Comparison of integral functionals depending on the second derivative of Chebyshev and Zolotarev polynomials,”
*East J. Approxim.*,**5**, No. 2, 151-182 (1999).zbMATHMathSciNetGoogle Scholar - 5.E. I. Zolotarev,
*Complete Collection of Works*[in Russian], Vol. 2, Izd. Akad. Nauk SSSR, Leningrad (1932).Google Scholar

## Copyright information

© Springer Science+Business Media, Inc. 2009