I will suggest 4 7 2 9 1 8 3 6 5
Read left to right for the lowest, and right to left for highest.
These meet the five conditions, but probably are not optimal. I suspect that those who use Mathematica have a built in function for this (minmax) -- it seems to have almost anything else one could think of -- , so will let them show their stuff. Clearly the 4 and the 5 must be on the extreme ends, since 45 is the sum of all the nine digits. We may exclude zero as one of the nine digits used, since 45 cannot be reached if that is substituted.
Edited on October 21, 2010, 8:21 pm