On general approach to Bessenrodt-Ono type inequalities and log-concavity property

In recent literature concerning integer partitions one can find many results related to both the Bessenrodt-Ono type inequalities and log-concavity property. 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 $F(\mathbb{N})\subset \mathbb{R}_{+}$, we have $F(a)F(b)>F(a+b)$ for sufficiently large positive integers $a, b$. Moreover, we show that if the sequence $(F(n))_{n\geq n_{0}}$ is log-concave and $\limsup_{n\rightarrow +\infty}F(n+n_{0})/F(n)<F(n_{0})$, then $F$ satisfies the Bessenrodt-Ono type inequality.