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

Let [Formula: see text] (resp. [Formula: see text]) denote the number of partitions of n with even (resp. odd) crank. Choi, Kang and Lovejoy established an asymptotic formula for [Formula: see text]. By utilizing this formula with the explicit bound, we show that [Formula: see text] for [Formula: see text] or 1 and [Formula: see text]. This result can be seen as the refinement of the classical result regarding the convexity of the partition function [Formula: see text], which counts the number of partitions of n. We also show that [Formula: see text] (resp. [Formula: see text]) is log-concave for [Formula: see text] and satisfies the higher order Turán inequalities for [Formula: see text] with the aid of the upper bound and the lower bound for [Formula: see text] and [Formula: see text].