Let $a_i$ and $b_i$, $i=1,\ldots,n$, be two finite sequences of numbers in $(0,1)$. We know that $a_i < b_i$ for all $i$. Is it then true that $a_1\cdots a_n < b_1\cdots b_n$?
If all the $a_i$ were the same, and all the $b_i$ were the same, this would be rather obvious since the functions $x^n$ for positive $n$ are known to be increasing on $(0,1)$. So in this special case, it works. But I'm not sure how to generalize this result.