  We consider differences of one- and two-variable finite products and provide
combinatorial proofs of the nonnegativity of certain coefficients. Since the
products may be interpreted as generating functions for certain integer
partitions, this amounts to showing a partition inequality. This extends
results due to Berkovich-Garvan and McLaughlin. We then apply the first
inequality and Andrews' Anti-telescoping Method to give a solution to an
'Ehrenpreis Problem' for recently conjectured sum-product identities of Kanade-
Russell. That is, we provide some evidence for Kanade-Russell's conjectures by
showing nonnegativity of coefficients in differences of product-sides as
Andrews-Baxter and Kadell did for the product sides of the Rogers-Ramanujan
identities.
