Certain positive q-series and inequalities for two-color partitions

We consider some q-series which depend on a pair of positive integers $(k,m)$. While positivity of these series holds for the first few values of $(k,m)$, the situation is quite unclear for other values of $(k,m)$. In addition, our series generate the number of certain two-color integer partitions weighted by (-1)ʲ where j is the number of even parts. Therefore, inequalities involving these partitions will be deduced from the positivity of their generating functions.