In the paper, we first survey some results on inequalities for bounding
harmonic numbers or Euler-Mascheroni constant, and then we establish a new
sharp double inequality for bounding harmonic numbers as follows: For
$n\in\mathbb{N}$, the double inequality
-\frac{1}{12n^2+{2(7-12\gamma)}/{(2\gamma-1)}}\le H(n)-\ln
n-\frac1{2n}-\gamma<-\frac{1}{12n^2+6/5} is valid, with equality in the
left-hand side only when $n=1$, where the scalars
$\frac{2(7-12\gamma)}{2\gamma-1}$ and $\frac65$ are the best possible.