Extremal properties of the derivatives of the Newman polynomials

Let $\Lambda _{n-1} := \{\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n}\}$be a set of $n$ distinct positive numbers. The span of

over ${\mathbb{R}}$ will be denoted by

Our main result of this note is the following.

Theorem. Suppose $0 < q \leq p \leq \infty$. Let $\mu$ be a non-negative integer. Then there are constants $c_{1}(p,q,\mu ) > 0$ and $c_{2}(p,q,\mu ) > 0$ depending only on $p$, $q$, and $\mu$ such that

where the lower bound holds for all $0 < q \leq p \leq \infty$ and for all $\mu \geq 0$, while the upper bound holds when $\mu = 0$ and $0 < q \leq p \leq \infty$ and when $\mu \geq 1$, $p \geq 1$, and $0 < q \leq p \leq \infty$.