Title: Polynomization of the Liu–Zhang inequality for the overpartition function
Abstract: Let $$\overline{p}(n)$$ denote the number of overpartitions of n. Liu and Zhang proved that $$\overline{p}(a) \overline{p}(b)>\overline{p}(a+b)$$ holds for any a, $$b>1$$ by using an analytic result of Engel. In this paper, we provide a combinatorial proof of the Liu–Zhang inequaity. More precisely, motivated by the polynomials $$P_{n}(x)$$ , which generalize the k-colored partition functions $$p_{-k}(n)$$ , we introduce the polynomials $$\overline{P}_{n}(x)$$ , which take the number of k-colored overpartitions of n as their special values. By utilizing combinatorial and analytic approaches, we prove that $$\overline{P}_{a}(x) \overline{P}_{b}(x)>\overline{P}_{a+b}(x)$$ holds for any a, $$b\ge 1$$ and real numbers $$x\ge 1$$ , except for $$(a,b,x)=(1,1,1),(2,1,1),(1,2,1)$$ .