Markov and Bernstein type inequalities for polynomials
For a real polynomial \(p(z)= \sum^n_{\nu =0} a_\nu z^\nu\) of degree \(n\) we have the inequality \[ \max_{-1\leq x\leq 1}|p'(x)|\leq n^2\max_{-1\leq x\leq 1}|p(x)| \] which is known as Markov's inequality. For a complex polynomial \(p(z)= \sum^n_{\nu= 0}c_\nu z^\nu\) of degree \(n\) we have \[ \max_{|z|= 1} |p'(z)|\leq n\max_{|z|=1} |p'(z)| \] which is known as Bernstein inequality. Both inequalities are the best possible and are attained for \(T_n(x)= \cos n\cos^{-1}x\) and \(p_n(z)= \lambda z^n\). These two inequalities have been the starting points for many considerations. In this paper, the authors present some of the research centered around these inequalities. The references include more then hundred papers.
scientific article; zbMATH DE number 1340269