Polynomization of the Liu-Zhang inequality for overpartition function

Let $\bar{p}(n)$ denote the overpartition function. Liu and Zhang showed that $\bar{p}(a)\bar{p}(b)>\bar{p}(a+b)$ 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 ${\bar{P}}_n(x)$, which take the number of $k$-colored overpartitions of $n$ as their special values. And by combining combinatorial and analytic approaches, we obtain that ${\bar{P}}_a(x){\bar{P}}_b(x)>{\bar{P}}_{a+b}(x)$ for all positive integers $a,b$ and real numbers $x\ge 1$ , except for $(a,b,x)=(1,1,1),(2,1,1),(1,2,1)$.