Inequalities for k-regular partitions

We build upon the work by Bessenrodt and Ono, as well as Beckwith and Bessenrodt concerning the combined additive and multiplicative behavior of the k-regular partition functions pₖ(n). Our focus is on addressing the solutions of the Bessenrodt--Ono inequality {equation*} p_k(a) \, p_k(b) > p_k(a+b). {equation*} We determine the sets Eₖ and Fₖ consisting of all pairs $(a,b)$, where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets E∞ and F∞ for the partition function $p(n)$. We prove by induction that Eₖ=E∞ and Fₖ=F∞ if and only if k ≥ 10. Beckwith and Bessenrodt used analytic methods to consider 2 ≤ k ≤ 6, while Alanazi, Gagola, and Munagi studied the case $k=2$ using combinatorial methods. Finally, we present a precise and comprehensive conjecture on the log-concavity of the k-regular partition function extending previous speculations by Craig and Pun. The case $k=2$ was recently proven by Dong and Ji.