$M_2$-Ranks of overpartitions modulo $6$ and $10$

In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let ${\bar{N}}_2(s,m,n)$ to be the number of overpartitions of $n$ whose $M_2$-rank is congruent to $s$ modulo $m$. For $M_2$-ranks modulo $3$, Lovejoy and Osburn derived the generating function of ${\bar{N}}_2(s,3,n)-{\bar{N}}_2(t,3,n)$, which implies the inequalities ${\bar{N}}_2(0,3,n)\ge {\bar{N}}_2(1,3,n)$. For $\mathrm{\ell }=6,10$, we consider the generating function ${\bar{R}}_{s,t}(d,\mathrm{\ell })$ of the $M_2$-rank differences $\overline{N}_2(s,\ell,\ell n/2+d) + \overline{N}_2(s+1,\ell,\ell n/2+d) - \overline{N}_2(t,\ell,\ell n/2+d) - \overline{N}2(t+1,\ell,\ell n/2+d)$\mathrm{.}BythemethodofLovejoyandOsburn,wederiveaformulafor$\overline{R}{0,2}(d,6)$\mathrm{.}Thisleadstotheinequalitiesfor$n\geq0$,$\overline{N}_2(0,6,3n)\geq\overline{N}_2(2,6,3n)$and$\overline{N}_2(0,6,3n+1) \geq \overline{N}_2(2,6,3n+1)$. Based on the valence formula for modular functions, we compute ${\bar{R}}_{0,4}(d,10)$ and ${\bar{R}}_{1,3}(d,10)$. In particular, we notice that the generating function ${\bar{R}}_{0,2}(2,6)$ can be expressed in terms of the third order mock theta function $\rho (q)$, and the generating functions ${\bar{R}}_{0,4}(4,10)$, ${\bar{R}}_{1,3}(1,10)$ and ${\bar{R}}_{1,3}(4,10)$ can also be expressed in terms of the tenth order mock theta functions $\varphi (q)$ and $\psi (q)$.