The Sharp Markov–Nikolskii Inequality for Algebraic Polynomials in the Spaces $L_q$ and $L_0$ on a Closed Interval

In this paper, an inequality between the $L_q$-mean of the $k$th derivative of an algebraic polynomial of degree $n\ge 1$ and the $L_0$-mean of the polynomial on a closed interval is obtained. Earlier, the author obtained the best constant in this inequality for $k=0$, $q\in[0,\infty]$ and $1\le k\le n$, $q\in\{0\}\cup[1,\infty]$. Here a new method for finding the best constant for all $0\le k\le n$, $q\in[0,\infty]$, and, in particular, for the case $1\le k\le n$, $q\in(0,1)$, which has not been studied before is proposed. We find the order of growth of the best constant with respect to $n$ as $n\to \infty$ for fixed $k$ and $q$.