Inequalities to give bounds on generalised harmonic numbers?

Let $n\geq1$ and $k\geq 2$. By the result of the paper http://dx.doi.org/10.1098/rspa.2017.0363, it holds that \begin{align*} H_n^{(k)} = \zeta (k) & + n^{ - k} \left( - \frac{n}{{k - 1}} + \frac{1}{2} - \sum\limits_{m = 1}^{M - 1} \frac{{B_{2m} }}{{(2m)!}}\frac{{\Gamma (k + 2m - 1)}}{{\Gamma (k)}}\frac{1}{{n^{2m - 1} }} \right. \\ & -\left. \theta _M (n,k)\frac{{B_{2M} }}{{(2M)!}}\frac{{\Gamma (k + 2M - 1)}}{{\Gamma (k)}}\frac{1}{{n^{2M - 1} }} \right), \end{align*} where $M\geq 1$, and $0<\theta _M (n,k)<1$ is an appropriate number. The $B_m$'s are the Bernoulli numbers. In particular, with $M=2$, $$ H_n^{(k)} < \zeta (k) + n^{ - k} \left( { - \frac{n}{{k - 1}} + \frac{1}{2} - \frac{k}{{12}}\frac{1}{n} + \frac{{k(k + 1)(k + 2)}}{{720}}\frac{1}{{n^3 }}} \right) $$ and $$ H_n^{(k)} > \zeta (k) + n^{ - k} \left( { - \frac{n}{{k - 1}} + \frac{1}{2} - \frac{k}{{12}}\frac{1}{n}} \right). $$ Note that the constant must be $1/2$ and not $\gamma$. It is also seen that for values of $k$ large enough, your upper bound is not valid.