On the Markov inequality in the L2L2-norm with the Gegenbauer weight

Let wλ(t)=(1−t2)λ−1/2wλ(t)=(1−t2)λ−1/2, λ>−1/2λ>−1/2, be the Gegenbauer weight function, and ‖⋅‖wλ‖⋅‖wλ denote the associated L2L2-norm, i.e., ‖f‖wλ:=(∫−11wλ(t)|f(t)|2dt)1/2. Denote by PnPn the set of algebraic polynomials of degree not exceeding nn. We study the best (i.e., the smallest) constant cn,λcn,λ in the Markov inequality ‖p′‖wλ≤cn,λ‖p‖wλ,p∈Pn, and prove that cn,λ<(n+1)(n+2λ+1)22λ+1,λ>−1/2.