Proof of two conjectures on the log-concavity of partition ranks and cranks

Let $M(m,n)$ and $N(m,n)$ denote the numbers of partitions of n with crank m and rank m, respectively. Ji and Zang proved that the crank sequence (M(m,n))|m|≤ n-1 is unimodal for every n≥44. Bringmann, Man, and Rolen subsequently established the unimodality of the rank sequence (N(m,n))|m|≤ n-2 for every n≥39. Ji and Zang also conjectured that (M(m,n))|m|≤ n-71 is log-concave for every n≥72, and that (N(m,n))|m|≤ n-72 is log-concave for every n≥73. In this paper, we prove both conjectures.