Sharp Markov brothers type inequality in the spaces $L_p$, $L_1$ on a closed interval

We study an inequality between the $L_p$-mean of the $(n-1)$th derivative of an algebraic polynomial of degree $n\geq2$ and the $L_1$-mean of this polynomial on a closed interval. Sharp constants and extremal polynomials are written for all $p\in[0,\infty]$.