Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [−1, 1] between the uniform norm and the norm of the space Lq(α,β), 1 ≤ q < ∞, with Jacobi weight ϕ(α,β)(x) = (1 − x)α(1 + x)β α ≥ β > −1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space Lq(α,β), 1 ≤ q < ∞, α>β≥−12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha > \beta \geqslant - \frac{1}{2}$$\end{document}, is attained.