On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties

Abstract In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function F of at most exponential growth satisfying the condition F (N) R+ F (N) ⊂ R +, we have F (a) F (b) >F (a+b) F (a) F (b) > F (a + b) for sufficiently large positive integers a, b. Moreover, we show that if the sequence (F (n) ) ₍ ₍_₀ (F (n) ) n ≥ n 0 is log-concave and ₍ + F (n+n₀) /F (n) lim sup n → + ∞ F (n + n 0) / F (n) F (n 0), then F satisfies the Bessenrodt–Ono type inequality.