Strict log-concavity of <i>k</i> -coloured partitions

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 $k\geq 2$ 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 $\boldsymbol{a},\boldsymbol{b}$ a , b of dollar sign n element of upper N dollar sign $n\in{\mathbb N}$ n ∈ N , if dollar sign b dollar sign $\boldsymbol b$ b majorizes dollar sign a dollar sign $\boldsymbol a$ a , then dollar sign pk left p