Rank and crank log-concavity conjectures

Let $M(m,n)$ and $N(m,n)$ denote respectively the number of partitions of $n$ with crank $m$ and rank $m$. Rank and crank log-concavity conjecture. The inequalities

$$M(m,n)^2\ge M(m-1,n)M(m+1,n)$$

hold for $n\ge71$ and $|m|\le n-71$, and

$$N(m,n)^2\ge N(m-1,n)N(m+1,n)$$

hold for $n\ge72$ and $|m|\le n-72$. These conjectures motivate the paper's generalization to Garvan $k$-rank statistics.