Higher Order Turan Inequalities for the Distinct Partition Function

We prove that the number $q(n)$ of partitions into distinct parts is log-concave for $n\ge 33$ and satisfies the higher order Turán inequalities for $n\ge 121$ conjectured by Craig and Pun. In doing so, we establish explicit error terms for $q(n)$ and for $q(n-1)q(n+1)\mathrm{/}q(n{)}^2$ based on Chern’s asymptotic formulas for $\eta$-quotients.