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$$φ_n(x) := \sqrt{n^{-2} + 1 - x_2,} .$$Namely,$$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n}\biggr)^2 - k.$$