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

  • V. D. Galan
  • I. A. Shevchuk

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, $$c(n, k) = \biggl( 1 + k \frac{\sqrt{ 1 + n^2} - 1}{n} \biggr)^2 - k.$$
Published
25.05.2017
How to Cite
GalanV. D., and ShevchukI. A. “Exact Constant in the Dzyadyk Inequality for the Derivative of an algebraic Polynomial”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 69, no. 5, May 2017, pp. 624-30, http://umj.imath.kiev.ua/index.php/umj/article/view/1721.
Section
Research articles