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