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

Abstract

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)|2wλ(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 ∥p′n∥wλ≤cn(λ)∥pn∥wλ,pn∈Pn, namely the constant cn(λ):=suppn∈Pn∥p′n∥wλ∥pn∥wλ. We derive explicit lower and upper bounds for the Markov constant cn(λ), which are valid for all n and λ.