Journal of Inequalities and Applications 
Volume 3 (1999), Issue 3, Pages 215-231
doi:10.1155/S1025583499000144

On a polynomial inequality of P. Erdős and T. Grünwald

D. Dryanov1 and Q. I. Rahman2

 1Department of Mathematics, University of Sofia, James Boucher 5, Sofia 1126, Bulgaria
2Département de Mathématiques et de Statistique, Université de Montréal, Montréal H3C 3J7, Canada
Received 15 May 1998; Revised 24 June 1998

Abstract

Let f be a polynomial with only real zeros having −1, +1 as consecutive zeros. It was proved by P. Erdős and T. Grünwald that if f(x)>0 on (−1,1), then the ratio of the area under the curve to the area of the tangential rectangle does not exceed 2/3. The main result of our paper is a multidimensional version of this result. First, we replace the class of polynomials considered by Erdős and Grünwald by the wider class ℭ consisting of functions of the form f(x):=(1−x2)ψ(x), where |ψ| is logarithmically concave on (−1,1), and show that their result holds for all functions in ℭ. More generally, we show that if f∈ℭ and max−1≤x≤1|f(x)|≤1, then for all p>0, the integral ∫−11|f(x)|pdx does not exceed ∫−11(1−x2)pdx. It is this result that is extended to higher dimensions. Our consideration of the class ℭ is crucial, since, unlike the narrower one of Erdős and Grünwald, its definition does not involve the distribution of zeros of its elements; besides, the notion of logarithmic concavity makes perfect sense for functions of several variables.