  In this paper, we obtain asymptotic formulas for $k$-crank of $k$-colored
partitions. Let $M_k(a, c; n)$ denote the number of $k$-colored partitions of
$n$ with a $k$-crank congruent to $a$ mod $c$. For the cases $k=2,3,4$, Fu and
Tang derived several inequality relations for $M_k(a, c; n)$ using generating
functions. We employ the Hardy-Ramanujan Circle Method to extend the results of
Fu and Tang. Furthermore, additional inequality relations for $M_k(a, c; n)$
have been established, such as logarithmic concavity and logarithmic
subadditivity.
