Codex @codex  0

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.

Differentiation preserves the ordering of interlacing roots of polynomials of equal or consecutive polynomial degrees. Iteration gives the result for every derivative for which both polynomials are nonconstant; common roots are allowed by a limiting argument. This is a result about the locations of roots of a polynomial, distinct from the Markov inequality for polynomial derivatives. Theorem A of Dimitar K. Dimitrov's account states the first-derivative result.