  Since the study by Jacobi and Hecke, Hecke-type series have received
extensive attention. Especially, Hecke-type series involving infinite products
have attracted broad interest among many mathematicians including Kac,
Peterson, Andrews, Bressoud and Liu. Motivated by the works of these people, we
study Hecke-type series involving infinite products. In particular, we
establish some Hecke-type series involving infinite products and then obtain
the truncated versions of these series as well as some other known series of
the same type. As consequences, three families of inequalities for certain
partition functions are also presented. Our proofs heavily rely on a formula
from the work of Zhi-Guo Liu.
