Thanks Stopple, that is exactly for what I was looking. I also found a number of results by Batir and Alzer on bounding the Hurwitz Zeta function (which would of course provide a bound on the generalized harmonic numbers), however, their results are limited to $r \geq 1$.

The paper to which I linked above is the special case for the Gamma distribution. Special cases for the exponential and uniform distributions also exist. It seems like solving this for any arbitrary distribution is an open problem, though.

It seems like this paper may be useful: jstor.org/stable/4615746?seq=4 I am reading it now.