Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions

We prove two monotonicity properties of N(m,n)N(m,n), the number of partitions of n with rank m. They are (i) for any nonnegative integers m and n,N(m,n)⩾N(m+2,n),N(m,n)⩾N(m+2,n), and, (ii) for any nonnegative integers m and n   such that n⩾12n⩾12, n≠m+2n≠m+2,N(m,n)⩾N(m,n−1).N(m,n)⩾N(m,n−1). G.E. Andrews, B. Kim, and the first author introduced ospt(n)ospt(n), a function counting the difference between the first positive rank and crank moments. They proved that ospt(n)>0ospt(n)>0. In another article, K. Bringmann and K. Mahlburg gave an asymptotic estimate for ospt(n)ospt(n). The two monotonicity properties for N(m,n)N(m,n) lead to stronger inequalities for ospt(n)ospt(n) that imply the asymptotic estimate.