Newman's inequality for Müntz polynomials on positive intervals
By D. J. Newman's inequality [J. Approximation Theory 18, 360-362 (1976; Zbl 0441.41002)], if \(0 \leq \lambda_0 < \lambda_1 < \dots\), then \[ {2 \over 3} \left( \sum^n_{j = 0} \lambda_j \right) |P |_{[0,1]} \leq |x P'(x) |_{[0,1]} \leq 11 \left( \sum^n_{j = 0} \lambda_j \right) |P |_{[0,1]} \] for any Müntz polynomial \(0 \neq P \in \text{Span} \{x^{\lambda_0}, \dots, x^{\lambda_n}\}\). In [Polynomials and polynomial inequalities (1995; Zbl 0840.26002)], the authors have shown that in the right-hand side inequality above \(|xP' (x) |_{[0,1]}\) can be replaced by \(|P' |_{[0,1]}\), whenever \(\lambda_0 = 0\) and \(\lambda_j \geq j\) for every \(j\). In this paper they prove that the same still holds if \(0 = \lambda_0 < \lambda_1 < \dots\) with \(\lambda_j \geq \delta_j\) for each \(j\), with some \(\delta > 0\), and if the line-segment \([0,1]\) is replaced by \([a,b] \subset (0, + \infty)\).
scientific article; zbMATH DE number 885476