Motivated by the conjecture of Sun on the log-convexity of \(\{\root n \of {p(n)}\}_{n\ge 60}\) , we obtain the log-convexity of \(\{\root n \of {spt(n)}\}_{n\ge 30}\) . For any real number \(\alpha \) , there exists an integer \(n(\alpha )\) such that the sequence \(\{\root n \of {spt(n)/n^{\alpha } } \}_{n\ge n(\alpha ) }\) is log-convex. Moreover, we establish an inequality on the ratio \({\root n-1 \of {spt(n-1)} }\big /{\root n \of {spt(n)}}\) by finding an upper bound of \(\Delta ^{2} \log \root n-1 \of {spt(n-1)}\) . In the end, we prove the higher order Turán inequalities for spt(n) conjectured by Chen by considering the reality of the Jensen polynomial \(J_{spt}^{d,n} (X)\) associated to spt(n).
