We prove inequalities between the first moment of the \(M_2\)-rank of overpartitions and the residual crank of such partitions. In order to obtain such inequalities, we need a result on the positivity of certain spt-cranks of overpartitions which were first introduced by Jennings-Shaffer.

中文翻译：

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$ M_2 $$ M2的第一刻之间的不等式-分区的曲柄和曲柄

我们证明了分区的\（M_2 \） -秩的第一时刻与此类分区的剩余曲柄之间的不等式。为了获得这样的不等式，我们需要得到詹宁斯-谢弗首次提出的某些隔板过高的曲柄的正性的结果。