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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* pₖ (a) \, pₖ (b) > pₖ (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.
Heim et al. (Sun,) studied this question.