Guo and Qi recently discovered sharp bounds for the harmonic numbers (*qq.v.* doi:10.1016/j.amc.2011.01.089). For example, they show that $$H_n < \ln(n) + \frac{1}{2n} + \gamma - \frac{1}{12n^2+\frac{6}{5}},$$ where $H_n$ is the $n$^{th} harmonic number and $\gamma$ is the Euler-Mascheroni constant.

**My question**: are there any similar bounds on the *generalized* harmonic numbers? By "generalized", I mean $H_{n,r}$ where $$H_{n,r} = \sum_{k=1}^n \frac{1}{k^r}.$$

I am specifically interested in the case when $r = \frac{1}{2}$.