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

In this paper, we obtain inequalities on $M_2$-ranks of overpartitions modulo $6$. Let $\overline{N}_2(s,m,n)$ 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 $\overline{N}_2(s,3,n)-\overline{N}_2(t,3,n)$, which implies the inequalities $\overline{N}_2(0,3,n)\geq\overline{N}_2(1,3,n)$. For $\ell=6, 10$, we consider the generating function $\overline{R}_{s,t}(d,\ell)$ of the $M_2$-rank differences $\overline{N}_2(s,\ell,\ell n/2+d) + \overline{N}_2(s+1,\ell,\ell n/2+d) - \overline{N}_2(t,\ell,\ell n/2+d) - \overline{N}_2(t+1,\ell,\ell n/2+d)$. By the method of Lovejoy and Osburn, we derive a formula for $\overline{R}_{0,2}(d,6)$. This leads to the inequalities for $n\geq0$, $\overline{N}_2(0,6,3n)\geq\overline{N}_2(2,6,3n)$ 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 $\overline{R}_{0,4}(d,10)$ and $\overline{R}_{1,3}(d,10)$. In particular, we notice that the generating function $\overline{R}_{0,2}(2,6)$ can be expressed in terms of the third order mock theta function $\rho(q)$, and the generating functions $\overline{R}_{0,4}(4,10)$, $\overline{R}_{1,3}(1,10)$ and $\overline{R}_{1,3}(4,10)$ can also be expressed in terms of the tenth order mock theta functions $\phi(q)$ and $\psi(q)$.