Ranks of overpartitions modulo 6 and 10

Lovejoy and Osburn proved formulas for the generating functions for the rank differences of overpartitions modulo 3 and 5. In this paper, we derive formulas for the generating functions for the rank differences of overpartitions modulo 6 and 10. With these generating functions, we obtain some equalities and inequalities on ranks of overpartitions modulo 6 and 10. We also relate these generating functions to the third order mock theta functions ω(q) and ρ(q) and the tenth order mock theta functions ϕ(q) and ψ(q).