2017
Том 69
№ 9

All Issues

Exact constant in the Dzyadyk inequality for the derivative of an algebraic polynomial

Galan V. D., Shevchuk I. A.


Abstract

For natural $k$ and $n \geq 2k$, we determine the exact constant $c(n, k)$ in the Dzyadyk inequality $$|| P^{\prime}_n\varphi^{1-k}_n ||_{C[ 1,1]} \leq c(n, k)n\| P_n\varphi^{-k}_n \|_{C[ 1,1]}$$ for the derivative $P^{\prime}_n$ of an algebraic polynomial $P_n$ of degree $\leq n$, where $$\varphi_n(x) := \sqrt{n^{-2} + 1 - x_2,} .$$ Namely, $$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n} \biggr)^2 - k.$$

Citation Example: Galan V. D., Shevchuk I. A. Exact constant in the Dzyadyk inequality for the derivative of an algebraic polynomial // Ukr. Mat. Zh. - 2017. - 69, № 5. - pp. 624-630.