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.
Categories: math.CA