The full Markov-Newman inequality for Müntz polynomials on positive intervals

Abstract:

For a function $f$ defined on an interval $[a,b]$ let \begin{equation*} \|f\|_{[a,b]} := \sup \{|f(x)|: x \in [a,b]\} . \end{equation*} The principal result of this paper is the following Markov-type inequality for Müntz polynomials.

Theorem. Let $n \geq 1$ be an integer. Let $\lambda _{0}$, $\lambda _{1}$, …, $\lambda _{n}$ be $n+1$ distinct real numbers. Let $0 < a < b$. Then \begin{align*} \frac {1}{3} \sum _{j=0}^{n}{|\lambda _{j}|} + \frac {1}{4\log (b/a)} (n-1)^{2} &\leq \sup _{0 \neq Q}{\frac {\|xQ^{\prime }(x)\|_{[a,b]}}{\|Q\|_{[a,b]}}}\\ &\leq 11 \sum _{j=0}^{n}{|\lambda _{j}|} + \frac {128}{\log (b/a)}(n+1)^2 , \end{align*} where the supremum is taken for all $Q \in \operatorname {span}\{x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots , x^{\lambda _{n}}\}$ (the span is the linear span over $\mathbb {R}$).