Inequalities for derivatives of polynomials with restricted zeros

Abstract:

Let $f$ be a polynomial of degree $n$ that has real roots all of which lie outside the interval $( - 1,1)$. P. Erdös proved that if $\left | f \right | \leqslant 1$ on this interval then $\left | {f’} \right | < ne/2$ on $[ - 1,1]$; this is much stronger than the results the inequalities of A. Markov and S. N. Bernstein would give. We will show that if only $n - k$ roots of $f$ are restricted as above, then $\left | {f’} \right | < {c_k}n$ holds on $[ - 1,1]$ with appropriate ${c_k}$. An upper estimate for the best ${c_k}$ is given. Results for higher derivatives and ${L^p}$ spaces are also obtained.