On the Markov inequality in the L2-norm with the Gegenbauer weight

Let wλ(t) := (1− t2)λ−1/2, where λ > − 12 , be the Gegenbauer weight function, let ‖ · ‖wλ be the associated L2-norm, ‖f‖wλ = {∫ 1 −1 |f(x)|wλ(x) dx }1/2 , and denote by Pn the space of algebraic polynomials of degree≤ n. We study the best constant cn(λ) in the Markov inequality in this norm ‖pn‖wλ ≤ cn(λ)‖pn‖wλ , pn ∈ Pn , namely the constant cn(λ) := sup pn∈Pn ‖pn‖wλ ‖pn‖wλ . We derive explicit lower and upper bounds for the Markov constant cn(λ), which are valid for all n and λ. MSC 2010: 41A17