Sharp Estimates of the Generalized Euler-Mascheroni Constant

Abstract

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)-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 $n\ge 1$.

6 pages