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Let $N(m,n)$ denote the number of partitions of $n$ with rank $m$, and let$M(m,n)$ denote the number of partitions of $n$ with crank $m$. Chan and Maoproved that for any nonnegative integers $m$ and $n$, $N(m,n)\geq N(m+2,n)$ andfor any nonnegative integers $m$ and $n$ such that $n\geq12$, $n\neq m+2$,$N(m,n)\geq N(m,n-1)$. Recently, Ji and Zang showed that for $n\geq 44$ and$1\leq m\leq n-1$, $M(m-1,n)\geq M(m,n)$ and for $n\geq 14$ and $0\leq m\leqn-2$, $M(m,n)\geq M(m,n-1)$. In this paper, we analogue the result of Ji andZang to overpartitions. Note that Bringmann, Lovejoy and Osburn introduced twotype of cranks on overpartitions, namely the first residue crank and the secondresidue crank. Consequently, for the first residue crank $\overline{M}(m,n)$,we show that $\overline{M}(m-1,n)\geq \overline{M}(m,n)$ for $m\geq 1$ and$n\geq 3$ and $\overline{M}(m,n)\geq \overline{M}(m,n+1)$ for $m\geq 0$ and$n\geq 1$. For the second residue crank $\overline{M2}(m,n)$, we show that$\overline{M2}(m-1,n)\geq \overline{M2}(m,n)$ for $m\geq 1$ and $n\geq 0$ and$\overline{M2}(m,n)\geq \overline{M2}(m,n+1)$ for $m\geq 0$ and $n\geq 1$.Moreover, let $M_k(m,n)$ denote the number of $k$-colored partitions of $n$with $k$-crank $m$, which was defined by Fu and Tang. They conjectured thatwhen $k\geq 2$, $M_k(m-1,n)\geq M_k(m,n)$ except for $k=2$ and $n=1$. With theaid of the inequality $\overline{M}(m-1,n)\geq \overline{M}(m,n)$ for $m\geq 1$and $n\geq 3$, we confirm this conjecture.

The Unimodality of the Crank on Overpartitions