Exact constant in the Dzyadyk inequality for the derivative of an algebraic polynomial - Galan, V. D.; Shevchuk, I. A. (Ukrains’kyi Matematychnyi Zhurnal, 2017)
For natural k and n ≥ 2k, we determine the exact constant c(n, k) in the Dzyadyk inequality || P^′_nφ^1-k_n ||_C[ 1,1] ≤ c(n, k)n\| P_nφ^-k_n \|_C[ 1,1] for the derivative P^′_n of an algebraic polynomial P_n of degree ≤ n, where φ_n(x) := n^-2 + 1 - x_2, . Namely, c(n, k) = ( 1 + k 1 + n^2 - 1n )^2 - k.