Markov [formula omitted] inequality with the Gegenbauer weight

Abstract

For the Gegenbauer weight function wλ(t)=(1−t2)λ−1∕2, λ>−1∕2, we denote by ‖⋅‖wλ the associated L2-norm, ‖f‖wλ≔(∫−11wλ(t)f2(t)dt)1∕2.We study the Markov inequality ‖p′‖wλ≤cn(λ)‖p‖wλ,p∈Pn,where Pn is the class of algebraic polynomials of degree not exceeding n. Upper and lower bounds for the best Markov constant cn(λ) are obtained, which are valid for all n∈N and λ>−12.