  In 2016 Bessenrodt--Ono discovered an inequality addressing additive and
multiplicative properties of the partition function. Generalization by several
authors have been given; on partitions with rank in a given residue class by
Hou--Jagadeesan and Males, on $k$-regular partitions by Beckwith--Bessenrodt,
on $k$-colored partitions by Chern, Fu, Tang, and Heim--Neuhauser on their
polynomization, and Dawsey--Masri on the Andrews ${\it spt}$-function. The
proofs depend on non-trivial asymptotic formulas related to the circle method
on one side, or a sophisticated combinatorial proof invented by
Alanazi--Gagola--Munagi. We offer in this paper a new proof of the
Bessenrodt--Ono inequality, which is built on a well-known recursion formula
for partition numbers. We extend the proof to the result of Chern--Fu--Tang and
its polynomization. Finally, we also obtain a new result.
