Speaker
Mikhail Komarov
(Vladimir State University)
Description
Let $\Pi_n$ be the class of algebraic polynomials $P$ of degree $n$, all of whose zeros lie on the segment $[-1,1]$.
In 1995, S.P. Zhou has proved the following Turán type
reverse Markov--Nikol'skii inequality:
$$\|P'\|_{L_p[-1,1]}>c\, {(\sqrt{n})}^{1-1/p+1/q}\, \|P\|_{L_q[-1,1]}, P\in \Pi_n,$$
where $ 0 < p \le q \le \infty, 1-1/p+1/q \ge 0 $ ($c>0$ is a constant independent of $P$ and $n$).
We show that Zhou's estimate remains true in the case $p=\infty$, $q>1$. Some of related Turán type inequalities are also discussed.
Primary author
Mikhail Komarov
(Vladimir State University)
