In this paper, we provide an estimate for approximating the generalized-Euler-constant function gamma (z)=sum_{k=1}^{infty }z ^{k-1} (frac{1}{k}-ln frac{k+1}{k} ) by its partial sum gamma _{N-1}(z) when 0< z<1. We obtain an asymptotic expansion for the generalized-Euler-constant function and show that the coefficients of the asymptotic expansion are explicitly expressed by the Eulerian fractions. Also, we find a recurrence relation for those coefficients. Using its relation with the generalized-Euler-constant function, we establish two inequalities for the generalized Somos’ quadratic recurrence constant. Moreover, two asymptotic expansions for the natural logarithm of the generalized Somos quadratic recurrence constant are presented.