Sharp Estimates of the Generalized Euler-Mascheroni Constant

TL;DR

This paper derives sharp bounds for the generalized Euler-Mascheroni constant using inequalities involving sequences and logarithmic functions, providing the best possible constants for these bounds.

Contribution

It determines the optimal constants in inequalities that tightly bound the difference between the sequences and the generalized Euler-Mascheroni constant.

Findings

Established the best possible constants for the inequalities.

Provided sharp bounds for the sequences related to the generalized Euler-Mascheroni constant.

Enhanced understanding of the approximation accuracy of these sequences.

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)-\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 $n\geq 1$.