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Let $M_0(n)$ (resp. $M_1(n)$) denote the number of partitions of $n$ witheven (reps. odd) crank. Choi, Kang and Lovejoy established an asymptoticformula for $M_0(n)-M_1(n)$. By utilizing this formula with the explicit bound,we show that $M_k(n-1)+M_k(n+1)>2M_k(n)$ for $k=0$ or $1$ and $n\geq 39$. Thisresult can be seen as the refinement of the classical result regarding theconvexity of the partition function $p(n)$, which counts the number ofpartitions of $n$. We also show that $M_0(n)$ (resp. $M_1(n)$) is log-concavefor $n\geq 94$ and satisfies the higher order Tur\'an inequalities for $n\geq207$ with the aid of the upper bound and the lower bound for $M_0(n)$ and$M_1(n)$.

Convexity and log-concavity of the partition function weighted by the parity of the crank