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

For a function $f$ defined on an interval $[a,b]$ let

The principal result of this paper is the following Markov-type inequality for Müntz polynomials. Theorem. Let $n \geq 1$ be an integer. Let $\lambda _{0}, \lambda _{1}, \ldots , \lambda _{n}$ be $n+1$ distinct real numbers. Let $0 < a < b$. Then

where the supremum is taken for all $Q \in \text{\rm span}\{x^{\lambda _{0}}, x^{\lambda _{1}}, \ldots , x^{\lambda _{n}}\}$ (the span is the linear span over ${\mathbb{R}}$).