  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.
