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

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.