On the Size of Polynomials with Curved Majorant Original Research Article

Let Cn(φ) denote all polynomials of degree n majorized by a positive C2 function φ on [−1,1], n = 0, 1, 2, ... . We establish that for every r ∈ (0, 1), there is an integer N(r, φ) > 0, such that, for all n ≥ N(r, φ), the polynomials in Cn(φ) could be as large as φ on [−r, r], i.e., [formula] for all x ∈ [−r, r] and n ≥ N(r, φ). This is related to a result of Newman and Rivlin [6].