  Based on work of Atkin and Swinnerton-Dyer on partition rank difference
functions, and more recent work of Lovejoy and Osburn, Mao has proved several
inequalities between partition ranks modulo $10$, and additional results modulo
$6$ and $10$ for the $M_2$ rank of partitions without repeated odd parts. Mao
conjectured some additional inequalities. We prove some of Mao's rank
inequality conjectures for both the rank and the $M_2$ rank modulo $10$ using
elementary methods.
