Markov-Duffin-Schaeffer inequality for polynomials with a circular majorant

Abstract:

If $p$ is a polynomial of degree at most $n$ such that $|p(x)| \leqslant \sqrt {1 - {x^2}}$ for $- 1 \leqslant x \leqslant 1$, then for each $k$, $\max |{p^{(k)}}(x)|$ on $[ - 1, 1]$ is maximized by the polynomial $({x^2} - 1){U_{n - 2}}(x)$ where ${U_m}$ is the $m$th Chebyshev polynomial of the second kind. The purpose of this paper is to investigate if it is enough to assume $|p(x)| \leqslant \sqrt {1 - {x^2}}$ at some appropriately chosen set of $n + 1$ points in $[ - 1, 1]$. The problem is inspired by a well-known extension of Markov’s inequality due to Duffin and Schaeffer.