The best bounds of harmonic sequence

For any natural number $n\in \mathbb{N}$, $ \frac{1}{2n+\frac1{1-\gamma}-2}\le \sum_{i=1}^n\frac1i-\ln n-\gamma<\frac{1}{2n+\frac13}, $where$\gamma=0.57721566490153286…m$denotesEule{r}^{\mathrm{\prime }}sconstant\mathrm{.}Theconstants$\frac{1}{1-\gamma}-2$and$\frac13$ are the best possible. As by-products, two double inequalities of the digamma and trigamma functions are established.