In recent years, there has been extensive work on inequalities among partition functions. In particular, Nicolas, and independently DeSalvo–Pak, proved that the partition function dollar sign p left parenthesis n right parenthesis dollar sign p (n) p (n) is eventually log-concave. Inspired by this and other results, Chern–Fu–Tang first conjectured the log-concavity of dollar sign k dollar sign k k -coloured partitions. Three of the authors and Tripp later proved this conjecture by introducing recursive sequences and a strict inequality for fractional partition functions, giving explicit errors. In this paper, we show that the log-concavity is, in fact, strict for dollar sign k greater than or equals 2 dollar sign k 2 k ≥ 2. We shed further light on this phenomenon by utilizing Hardy–Littlewood–Pólya’s notion of majorizing. We prove that for partitions dollar sign a comma b dollar sign a, b a, b of dollar sign n element of upper N dollar sign n N n ∈ N, if dollar sign b dollar sign b b majorizes dollar sign a dollar sign a a, then dollar sign pk left p
Bringmann et al. (Mon,) studied this question.