Sharp Estimates of the Generalized Euler-Mascheroni Constant

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 $n\geq 1$.