Sharp Estimates of the Generalized Euler-Mascheroni Constant
Let $a\in (0, \infty)$, $γ(a)$ 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 $α_i, β_i (i=1,2,3,4)$ such that the following inequalities \begin{align*} \frac{1}{2(n+a)-α_1}\leq &γ(a)-x_n< \frac{1}{2(n+a)-β_1},\\ \frac{1}{2(n+a)-α_2}\leq &y_n-γ(a)< \frac{1}{2(n+a)-β_2},\\ \frac{1}{2(n+a)}+\frac{α_3}{(n+a)^2}\leq &γ(a)-x_n<\frac{1}{2(n+a)}+\frac{β_3}{(n+a)^2},\\ \frac{1}{2(n+a-1)}+\frac{α_4}{(n+a-1)^2}< &y_n-γ(a)\leq\frac{1}{2(n+a-1)}+\frac{β_4}{(n+a-1)^2}. \end{align*} are valid for all integers $n\geq 1$.
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