Asymptotics and inequalities for the distinct partition function

In this paper, we give explicit error bounds for the asymptotic expansion of the shifted distinct partition function $q(n+s)$ for any nonnegative integer $s$. Then based on this refined asymptotic formula, we give the exact thresholds of $n$ for the inequalities derived from the invariants of the quartic binary form, the double Turán inequalities, the Laguerre inequalities and their corresponding companion versions.