A Markov type inequality for higher derivatives of polynomials (Q749794)

Let \(\| \cdot \|\) be the weighted \(L^ 2\)-norm with Laguerre weight \(w(t)=\exp (-t)\). Let \(P_ n\) be the set of all complex polynomials whose degree does not exceed n and let \(\gamma_ n^{(r)}:=\sup_{f\in P_ n}(\| f^{(r)}\| /\| f\|).\) In this paper upper and lower bounds for \(\gamma_ n^{(r)}\) are given. Its asymptotic behaviour as \(n\to \infty\) is investigated. Moreover, some identities concerning arbitrary orthogonal polynomials are derived.