  Many papers have studied inequalities for partition functions. Recently, a
number of papers have considered mixtures between additive and multiplicative
behavior in such inequalities. In particular, Chern-Fu-Tang and Heim-Neuhauser
gave conjectures on inequalities for coefficients of powers of the generating
partition function. These conjectures were posed in the context of colored
partitions and the Nekrasov-Okounkov formula. Here, we study the precise size
of differences of products of two such coefficients. This allows us to prove
the Chern-Fu-Tang conjecture and to show the Heim-Neuhauser conjecture in a
certain range. The explicit error terms provided will also be useful in the
future study of partition inequalities. These are laid out in a user-friendly
way for the researcher in combinatorics interested in such analytic questions.
