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\begin{document}$$a\in (0, \infty )$$\end{document} and s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s \in (0, 1)$$\end{document}, and let γ\documentclass[12pt]{minimal}
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\begin{document}$$\gamma $$\end{document} denote the Euler-Mascheroni constant. The sequence {yn(a,s)}n∈N\documentclass[12pt]{minimal}
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\begin{document}$$\{y_n(a, s)\}_{n\in \mathbb {N}}$$\end{document} is defined by yn(a,s)=1as+1(a+1)s+⋯+1(a+n-1)s-(a+n-1)1-s-a1-s1-s\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} y_n(a, s)&=\frac{1}{a^s}+\frac{1}{(a+1)^s}+\cdots +\frac{1}{(a+n-1)^s}-\frac{(a+n-1)^{1-s}-a^{1-s}}{1-s} \end{aligned}$$\end{document}for each n∈N:={1,2,3,…}\documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}:=\{1,2,3,\ldots \}$$\end{document}. The sequence {yn(a,s)}n∈N\documentclass[12pt]{minimal}
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\begin{document}$$\{y_n(a, s)\}_{n\in \mathbb {N}}$$\end{document} is convergent and its limit, denoted by ℓ(a,s)\documentclass[12pt]{minimal}
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\begin{document}$$\ell (a, s)$$\end{document}, is a generalized Euler-Mascheroni constant. In this paper, we consider complete asymptotic expansions and inequalities related to generalized Euler-Mascheroni constant ℓ(a,s)\documentclass[12pt]{minimal}
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\begin{document}$$\ell (a,s)$$\end{document}. We determine the coefficients aj\documentclass[12pt]{minimal}
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\begin{document}$$a_j$$\end{document} and bj\documentclass[12pt]{minimal}
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\begin{document}$$b_j$$\end{document} such that ∑k=1n1k-lnnp+∑j=1pajnp-jnq+∑j=1qbjnq-j=γ+O1np+q+1,n→∞,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \sum _{k=1}^{n}\frac{1}{k}-\ln \frac{n^p+\sum _{j=1}^{p}a_jn^{p-j}}{n^q+\sum _{j=1}^{q}b_jn^{q-j}}=\gamma +O\left( \frac{1}{n^{p+q+1}} \right) ,\qquad n\rightarrow \infty , \end{aligned}$$\end{document}where p≥1\documentclass[12pt]{minimal}
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\begin{document}$$p\ge 1$$\end{document} and q≥0\documentclass[12pt]{minimal}
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\begin{document}$$q\ge 0$$\end{document} are any given integers and p=q+1\documentclass[12pt]{minimal}
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\begin{document}$$p=q+1$$\end{document}. This solves an open problem of Mortici.
