As a consequence, we derive several inequalities for the partition function $t(n)$ counting the partition triples
$(\lambda, \mu,\nu)$ of $n$ such that the total sum of parts of $\lambda$, $\mu$ and $\nu$ is $n$.
We also reprove the conjecture made by Guo-Zeng.
Moreover, we give some the extensions of  truncated theorems obtained by Andrews, Merca, Wang and Yee.
Our proofs heavily rely on the theory of Bailey pairs.
In this talk, we will also introduce some inequalities for the overpartition function.
Let $\overline{p}(n)$ denote the overpartition funtion.
Engel showed that for $n\geq2$, $\overline{p}(n)$ satisfied the Tur\'{a}n inequalities,
that is, $\overline{p}(n)^2-\overline{p}(n-1)\overline{p}(n+1)>0$ for $n\geq2$.
In this talk, we prove several inequalities for $\overline{p}(n)$.
Moreover, motivated by the work of Chen, Jia and Wang,
we find that the higher order Tur\'{a}n inequalities of $\overline{p}(n)$ can also be determined.