  We consider some multivariate rational functions which have (or are
conjectured to have) only positive coefficients in their series expansion. We
consider an operator that preserves positivity of series coefficients, and
apply the inverse of this operator to the rational functions. We obtain new
rational functions which seem to have only positive coefficients, whose
positivity would imply positivity of the original series, and which, in a
certain sense, cannot be improved any further.
