Sharp bounds for harmonic numbers

TL;DR

This paper surveys existing inequalities for harmonic numbers and introduces a new sharp double inequality that tightly bounds harmonic numbers with optimal constants.

Contribution

The paper provides the first sharp double inequality for harmonic numbers with best possible constants, improving bounds and understanding of their behavior.

Findings

Established a new sharp double inequality for harmonic numbers.

Identified the best possible constants for the bounds.

Provided a detailed survey of existing inequalities.

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(7-12\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.