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 $$F(\mathbb {N})\subset \mathbb {R}_{+}$$ 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 $$(F(n))_{n\ge n_{0}}$$ ( F ( n ) ) n ≥ n 0 is log-concave and $$\limsup _{n\rightarrow +\infty }F(n+n_{0})/F(n)<F(n_{0})$$ lim sup n → + ∞ F ( n + n 0 ) / F ( n ) < F ( n 0 ) , then F satisfies the Bessenrodt–Ono type inequality.