Journal of Inequalities and Applications
Volume 3 (1999), Issue 4, Pages 349-387

Markov and Bernstein type inequalities for polynomials

N. K. Govil1 and R. N. Mohapatra2

1Department of Mathematics, Auburn University, Auburn 36849, AL, USA
2Department of Mathematics, University of Florida, Orlando 32816, FL, USA

Received 30 July 1998; Revised 1 September 1998

Copyright © 1999 N. K. Govil and R. N. Mohapatra. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In an answer to a question raised by chemist Mendeleev, A. Markov proved that if p(z)=v=0navzv is a real polynomial of degree n, then max1x1|p(x)|n2max1x1|p(x)|. The above inequality which is known as Markov’s Inequality is best possible and becomes equality for the Chebyshev polynomial Tn(x)=cosncos1x.

Few years later, Serge Bernstein needed the analogue of this result for the unit disk in the complex plane instead of the interval [1,1] and the following is known as Bernstein’s Inequality.

If p(z)=v=0navzv is a polynomial of degree n then max|z|=1|p(z)|nmax|z|=1|p(z)|. This inequality is also best possible and is attained for p(z)=λzn, λ being a complex number.

The above two inequalities have been the starting point of a considerable literature in Mathematics and in this article we discuss some of the research centered around these inequalities.