On Certain Mean Values of Polynomials on the Unit Interval

Abstract

For any continuous function f:[−1, 1]↦C and any p∈(0, ∞), let ‖f‖p≔(2−1∫1−1|f(x)|pdx)1/p; in addition, let ‖f‖∞≔max−1⩽x⩽1|f(x)|. It is known that if f is a polynomial of degree n, then for all p>0,‖f‖∞⩽Cpn2/p‖f‖p,where Cp is a constant depending on p but not on n. In this result of Nikolskiı (1951), which was independently obtained by Szegö and Zygmund (1954), the order of magnitude of the bound is the best possible. We obtain a sharp version of this inequality for polynomials not vanishing in the open unit disk. As an application we prove the following result. If f is a real polynomial of degree n such that f(−1)=f(1)=0 and f(z)≠0 in the open unit disk, then for p>0 the quantity ‖f′‖∞/‖f‖p is maximized by polynomials of the form c(1+x)n−1(1−x), c(1+x)(1−x)n−1, where c∈R\\{0}. This extends an inequality of Erdős (1940).