On rational approximation to e

We prove: The inequality \e - p/q\ greater than or equal to c(1) log log q/q(2) log q holds for all positive integers p and q with g greater than or equal to 2, if and only if c(1) less than or equal to 0.386249199819.... And, the reverse inequality \e - p/q\ < c(2) log log q/q(2) log q has infinitely many solutions in integers p, q, if and only if c(2) greater than or equal to 1/2. (C) 1998 Academic Press.