Duffin-Schaeffer polynomial derivative inequality ($\Vert p^{(k)}{\Vert }_{\infty }\le T_n^{(k)}(1)$)

If a real degree-at-most-$n$ polynomial has absolute value at most one at every extremum $\cos (j\pi /n)$ of the Chebyshev polynomial $T_n$, its $k$th derivative on $[-1,1]$ is bounded by $T_n^{(k)}(1)$, for $1\le k\le n$. This strengthens the Markov inequality for polynomial derivatives by replacing a bound on the whole interval with these finitely many samples. The node set is rigid, as quantified by uniqueness of Chebyshev derivative-norming nodes.

For $n\ge 1$, any distinct node set in $[-1,1]$ different from the $n+1$ Chebyshev polynomial extrema admits one degree-$n$ polynomial $q$, bounded by one at all those nodes, whose every nonconstant derivative violates the corresponding sharp endpoint bound. In decreasing order set $q(t_i)=(-1{)}^i$. The endpoint derivative signs of Lagrange cardinal polynomials give $q^{(k)}(1)={\sum }_i|L_i^{(k)}(1)|$. Interpolating $T_n$ on the same nodes showsEach term is nonnegative and at least one is positive, because the other node set includes a nonextremal point. Uniqueness is a claim about node sets: permuting the same nodes does not change their norming property.