On a polynomial inequality of P. Erdos and T. Grunwald

Let f be a polynomial with only real zeros having -1, +1 as consecutive zer os. It was proved by P. Erdos and T. Grunwald that if f(x) > 0 on (-1, I), then the ratio of the area under the curve to the area of the tangential re ctangle does not exceed 2/3. The main result of our paper is a multidimensi onal version of this result. First, we replace the class of polynomials con sidered by Erdos and Grunwald by the wider class C consisting of functions of the form f(x) := (1-x(2))psi(x), where \psi\ is logarithmically concave on (-1, 1), and show that their result holds for all functions in C. More g enerally, we show that if f is an element of C and max(-1) (less than or eq ual to x less than or equal to 1)\f(x)\ less than or equal to 1, then for a ll p > 0, the integral integral(-1)(1) \ f(x)\(P) dx does not exceed integr al(-1)(1) (1 - x(2))(P) dx. It is this result that is extended to higher di mensions. Our consideration of the class C is crucial, since, unlike the na rrower one of Erdos and Grunwald, its definition does not involve the distr ibution of zeros of its elements; besides, the notion of logarithmic concav ity makes perfect sense for functions of several variables.