An Inequality of the Markov–Bernstein Type for Polynomials

Abstract

Let $ - \infty \leq a < b \leq \infty $ and denote by $w:(a,b) \to \mathbb{R}$ a positive and integrable function, with all moments \[ \int_a^b {t^n } w(t)dt\] finite. For any polynomial f with complex coefficients, we write \[ \| f \| = \left\{ {\int_a^b {| {f(t)} |^2 w(t)dt} } \right\}^{{1 / 2}} .\] Then there exists a constant $\gamma _n $ (depending on $a,b,w$ but not on f) such that, for every polynomial f with complex coefficients and of degree $ \leq n$, \[ \| {f'} \| \leq \gamma _n \| f \|.\] An admissible value for $\gamma _n $ can be expressed very simply in terms of the system of orthonormal polynomials associated with the interval $(a,b)$ and the function w.