  We prove a generalized Faulhaber inequality to bound the sums of the $j$-th
powers of the first $n$ (possibly shifted) natural numbers. With the help of
this inequality we are able to improve the known bounds for bracketing numbers
of $d$-dimensional axis-parallel boxes anchored in $0$ (or, put differently, of
lower left orthants intersected with the $d$-dimensional unit cube $[0,1]^d$).
We use these improved bracketing numbers to establish new bounds for the
star-discrepancy of negatively dependent random point sets and its expectation.
We apply our findings also to the weighted star-discrepancy.
