Some sharp inequalities for algebraic polynomials (Q1273406)

I. K. Daugavet, S. Z. Rafalson, and S. V. Konjagin [see \textit{G. V. Milovanović, D. S. Mitrinović} and \textit{Th. M. Rassias} [Topics in polynomials: extremal problems, inequalities, zeroes (1994; Zbl 0848.26001), p. 595-597)] considered the following general extremal problem for polynomials in different norms with Gegenbauer weights \[ A_{n,k}(r,\mu;p,\nu)=\sup _{P\in \mathcal P _n} \frac{\| P^{(k)}\| _{p,\nu}}{\| P\| _{r,\mu }}, \] where \(\mathcal P _n\) is the set of real algebraic polynomials of degree at most \(n\) and \[ \| f\| _{p,\nu }=\left( \int_{-1}^1 | f(t)(1-t^2)^{\nu }| ^p dt\right)^{1/p}, \qquad 0\leq p < +\infty. \] The author finds explicit expressions for \(A_{n,0}(2,3;2,1)\) and \(A_{n,0}(2,1;2,3).\) An analogous problem is considered for Jacobi weights \((1-t)^{\alpha }(1+t)^\beta\), \(| \alpha | =| \beta | =1\) and explicit expressions are found in several cases.