Markov-Bernstein type inequalities for constrained polynomials with real versus complex coefficients Academic Article uri icon

abstract

  • Let Pcn,k denote the set of all polynomials of degree at most n with complex coefficients and with at most k (0 ≤ k ≤ n) zeros in the open unit disk. Let Pn,k denote the set of all polynomials of degree at most n with real coefficients and with at most k (0 ≤ k ≤ n) zeros in the open unit disk. Associated with 0 ≤ k ≤ n and cursive Greek chi ∈ [-1, 1], let B*n,k,cursive Greek chi := max {√n(k + 1)/1 - cursive Greek chi2, nlog (e/1 - cursive Greek chi2)}, Bn,k,cursive Greek chi := √n(k + 1)/1 - cursive Greek chi2, and M*n,k := max{n(K + 1), nlogn}, Mn,k := n(k + 1). It is shown that c1 min{B*n,k,cursive Greek chi, M*n,k} ≤ supp∈Pcn,k |p′(cursive Greek chi)|/∥p∥[-1,1] ≤ c2 min{B*n,k,cursive Greek chi, M*n,k} for every cursive Greek chi ∈ [-1,1], where c1 > 0 and c2 > 0 are absolute constants. Here ∥ · ∥[-1,1] denotes the supremum norm on [-1,1]. This result should be compared with the inequalities c3 min{Bn,k,cursive Greek chi, Mn,k} ≤ supp∈Pn,k |p′(cursive Greek chi)|/∥p∥[-1,1] ≤ c4 min{Bn,k,cursive Greek chi, Mn,k}, for every cursive Greek chi ∈ [-1,1], where c3 > 0 and c4 > 0 are absolute constants. The upper bound of this second result is also fairly recent; and it may be surprising that there is a significant difference between the real and complex cases as far as Markov-Bernstein type inequalities are concerned. The lower bound of the second result is proved in this paper. It is the final piece in a long series of papers on this topic by a number of authors starting with Erdös in 1940.

author list (cited authors)

  • Erdélyi, T.

citation count

  • 4

publication date

  • December 1998