Sharp bounds for harmonic numbers

Abstract

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(7-12γ)}/{(2γ-1)}}\le H(n)-\ln n-\frac1{2n}-γ<-\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 $\frac{6}{5}$ are the best possible.

7 pages

References in corpus (1)

• The best bounds of harmonic sequence

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