Markov-type estimates for certain classes of constrained polynomials
Let \(m,n\in {\mathbb{N}}\), \(k\in {\mathbb{N}}_ 0\) and suppose that \(0\leq k\leq n\) and \(0<r\leq 1\); let \(D_ 1(r)^{\pm}=\{z\in {\mathbb{C}}:\) \(| z\pm (1- r)| <r\}\), denote by \(S^ k_ n(r)^{\pm}\) the set of all those real algebraic polynomials of degree \(\leq n\) and with at most k roots in \(D_ 1(r)^{\pm}\), and put \(S^ k_ n(r)=S^ k_ n(r)^+\cap S^ k_ n(r)^-.\) The author proves that for all \(s\in S^ k_ n(r)\), \(\| s^{(m)}\| \leq c_ 1(m)(n(k+1)^ 2/\sqrt{r})^ m\| s\|,\) where \(\| \cdot \|\) denotes the supremum norm over [-1,1]. He also shows that if \(n^{-2}\leq r\leq 1\), \(k=0\) and \(1\leq m\leq n\), then up to the constant \(c_ 1(m)\), this result is sharp.
scientific article; zbMATH DE number 4138489