Polynomization of the Liu-Zhang inequality for overpartition function

Let $\overline{p}(n)$ denote the overpartition function. Liu and Zhang showed that $\overline{p}(a) \overline{p}(b)>\overline{p}(a+b)$ for all integers $a,b>1$ by using an analytic result of Engle. We offer in this paper a combinatorial proof to the Liu-Zhang inequaity. More precisely, motivated by the polynomials $P_{n}(x)$ , which generalize the $k$-colored partitions function $p_{-k}(n)$, we introduce the polynomials $\overline{P}_{n}(x)$, which take the number of $k$-colored overpartitions of $n$ as their special values. And by combining combinatorial and analytic approaches, we obtain that $\overline{P}_{a}(x) \overline{P}_{b}(x)>\overline{P}_{a+b}(x)$ for all positive integers $a,b$ and real numbers $x \ge 1$ , except for $(a,b,x)=(1,1,1),(2,1,1),(1,2,1)$.