Hermite-Pade approximants to exponential functions and an inequality of Mahler

We improve Mahler's inequality \e(g) - a\ > g(-33g), a is an element of N, where g is any sufficiently large positive integer by decreasing the consta nt 33 to 19.183. This we do by computing precise asymptotics for a set of a pproximants to the exponential which is slightly different from the classic al Hermite-Pade: approximants. These approximants are related to the Legend re-type polynomials studied by Hata, which allows us to use his results abo ut the arithmetic of the coefficients. (C) 1999 Academic Press.