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 \(p_k(n)\) . Our focus is on addressing the solutions of the Bessenrodt–Ono inequality \(\begin{aligned} p_k(a) \, p_k(b) > p_k(a+b). \end{aligned}\) We determine the sets \(E_k\) and \(F_k\) consisting of all pairs (a, b), where we have equality or the opposite inequality. Bessenrodt and Ono previously determined the exception sets \(E_{\infty }\) and \(F_{\infty }\) for the partition function p(n). We prove by induction that \(E_k=E_{\infty }\) and \(F_k=F_{\infty }\) if and only if \(k \ge 10\) . Beckwith and Bessenrodt used analytic methods to consider \(2 \le k \le 6\) , 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.
