  Let $a\in (0, \infty)$, $\gamma(a)$ be the Generalized Euler-Mascheroni
Constant, and let \begin{align*}
&x_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n}{a},\\
&y_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n-1}{a}.
\end{align*} In this paper, we determine the best possible constants $\alpha_i,
\beta_i (i=1,2,3,4)$ such that the following inequalities \begin{align*}
\frac{1}{2(n+a)-\alpha_1}\leq &\gamma(a)-x_n< \frac{1}{2(n+a)-\beta_1},\\
\frac{1}{2(n+a)-\alpha_2}\leq &y_n-\gamma(a)< \frac{1}{2(n+a)-\beta_2},\\
\frac{1}{2(n+a)}+\frac{\alpha_3}{(n+a)^2}\leq
&\gamma(a)-x_n<\frac{1}{2(n+a)}+\frac{\beta_3}{(n+a)^2},\\
\frac{1}{2(n+a-1)}+\frac{\alpha_4}{(n+a-1)^2}<
&y_n-\gamma(a)\leq\frac{1}{2(n+a-1)}+\frac{\beta_4}{(n+a-1)^2}. \end{align*}
are valid for all integers $n\geq 1$.
