Inequalities for polynomials on the unit interval

Abstract:

Let ${p_n}(z) = \sum \nolimits _{k = 0}^n {{a_k}{z^k}}$ be a polynomial of degree at most n with real coefficients. Generalizing certain results of I. Schur related to the well-known inequalities of Chebyshev and Markov we prove that if ${p_n}(z)$ has at most $n - 1$ distinct zeros in $( - 1,1)$, then \[ \begin {array}{*{20}{c}} {|{a_n}| \leqslant {2^{n - 1}}{{\left ( {\cos \frac {\pi }{{4n}}} \right )}^{2n}}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|,} \\ {\max \limits _{ - 1 \leqslant x \leqslant 1} |{{p’}_n}(x)| \leqslant {{\left ( {n\cos \frac {\pi }{{4n}}} \right )}^2}\max \limits _{ - 1 \leqslant x \leqslant 1} |{p_n}(x)|.} \\ \end {array} \]