In this talk, I will prove several inequalities for $\overline{p}(n)$.

Moreover, the higher order Tur\'{a}n inequalities of $\overline{p}(n)$ can also be determined. After that, I will establish the generating functions for the ranks differences of overpartitions modulo $6$ and $10$, and obtain some inequalities between the ranks of overpartitions modulo $6$ and $10$. In light of these generating functions, some relations between mock theta functions and the ranks of overpartitions modulo $6$ and $10$ can also be found.