On certain mean values of polynomials on the unit interval

For any continuous function f:[--1, 1] --> C and any p is an element of(0, infinity), let parallel to f parallel to(p):= (2(-1) integral(-1)(1)\f(x)\( p) dx)(1/p); in addition, let parallel to f parallel to(infinity):= max(-1 less than or equal to x less than or equal to 1) \f(x)\. It is known that i f f is a polynomial of degree n, then for all p > 0, parallel to f parallel to(infinity) less than or equal to C(p)n(2/p) parall el to f parallel to(p), where C-p is a constant depending on p but not on n. In this result of Niko lskii (1951), which was independently obtained by Szego and Zygmund (1954), the order of magnitude of the bound is the best possible, We obtain a shar p version of this inequality far polynomials not vanishing in the open unit disk. As an application we prove the following result. If f is a real poly nomial of degree It such that f(-1)=f(1)=0 and f(z)not equal 0 in the open unit disk, then for p>0 the quantity parallel to f'parallel to(infinity)/pa rallel to f parallel to(p) is maximized by polynomials of the form c(1 + x) (n-1) (1 - x), c(1 + x)(1 - x)(n-1), where c is an element of R\{0}. This e xtends an inequality of Erdos (1940). (C) 1999 Academic Press.