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Proof of a Limited Version of Mao's Partition Rank Inequality using a Theta Function Identity

Ramanujan's congruence $p(5k+4) \equiv 0 \pmod 5$ led Dyson \cite{dyson} to conjecture the existence of a measure "rank" such that $p(5k+4)$ partitions of $5k+4$ could be divided into sub-classes with equal cardinality to give a direct proof of Ramanujan's congruence. The notion of rank was extended to rank differences by Atkin and Swinnerton-Dyer \cite{atkin}, who proved Dyson's conjecture. More recently, Mao proved several equalities and inequalities, leaving some as conjectures, for rank differences for partitions modulo 10 \cite{mao10} and for $M_2$ rank differences for partitions with no repeated odd parts modulo $6$ and $10$ \cite{maom2}. Alwaise et. al. proved four of Mao's conjectured inequalities \cite{swisher}, while leaving three open. Here, we prove a limited version of one of the inequalities conjectured by Mao.