Some weighted polynomial inequalities in \(L^ 2\)-norm (Q1335047)

It is established a unified inequality describing the estimate of the norm of polynomial derivative via the norm of polynomials itself in the weighted \(L_ 2\)-norm, which characterizes the classical orthogonal Hermite polynomials (on the interval \((-\infty,\infty)\)), generalized Laguerre polynomials (on the interval \((0,-\infty)\)) and Jacobi polynomials (on the segment \([-1,1]\)). In the special case of the general inequality for each of this classical polynomials the corresponding weights and best possible constants are investigated. The obtained result gives a new characterization of the classical orthogonal polynomials by extremal properties in weighted inequalities.