On Markov-Duffin-Schaeffer inequalities with a majorant. II

TL;DR

This paper extends the study of Markov inequalities with a majorant, proving that the snake-polynomial attains the upper bound for derivatives even in non-symmetric cases with positive Chebyshev expansion.

Contribution

It proves the conjecture that the snake-polynomial attains the upper bound for derivatives in non-symmetric majorants with positive Chebyshev expansion.

Findings

The conjecture holds for non-symmetric majorants with positive Chebyshev expansion.

The snake-polynomial attains the upper bound in broader cases than previously proven.

The results generalize earlier symmetric majorant cases.

Abstract

We are continuing out studies of the so-called Markov inequalities with a majorant. Inequalities of this type provide a bound for the $k$-th derivative of an algebraic polynomial when the latter is bounded by a certain curved majorant $\mu$. A conjecture is that the upper bound is attained by the so-called snake-polynomial which oscillates most between $\pm \mu$, but it turned out to be a rather difficult question. In the previous paper, we proved that this is true in the case of symmetric majorant provided the snake-polynomial has a positive Chebyshev expansion. In this paper, we show that that the conjecture is valid under the condition of positive expansion only, hence for non-symmetric majorants as well.