Inequalities between overpartition ranks for all moduli

In this paper we give a full description of the inequalities that can occur between overpartition ranks. If $\overline{N}(a,c,n)$ denotes the number of overpartitions of $n$ with rank congruent to $a$ modulo $c,$ we prove that for any $c\ge7$ and $0\le a<b\le\left\lfloor\frac{c}{2}\right\rfloor$ we have $\overline{N}(a,c,n)>\overline{N}(b,c,n)$ for $n$ large enough. That the sign of the rank differences $\overline{N}(a,c,n)-\overline{N}(b,c,n)$ depends on the residue class of $n$ modulo $c$ in the case of small moduli, such as $c=6,$ is known due to the work of Ji, Zhang and Zhao (2018) and Ciolan (2020). We show that the same behavior holds for $c\in\{2,3, 4,5\}.$