  Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positive
integers and let $k\in\mathbb{N}_+$ be fixed. The function $p_\mathcal{A}(n,k)$
counts the number of partitions of $n$ with parts in the multiset
$\{a_1,a_2,\ldots,a_k\}$. We find out a new type of Bessenrodt-Ono inequality
for the function $p_\mathcal{A}(n,k)$. Further, we discover when and under what
conditions on $k$, $\{a_1,a_2,\ldots,a_k\}$ and $N\in\mathbb{N}_+$, the
sequence $\left(p_\mathcal{A}(n,k)\right)_{n=N}^\infty$ is log-concave. Our
proofs are based on the asymptotic behavior of $p_\mathcal{A}(n,k)$, in
particular, we apply the results of Netto and P\'olya-Szeg\"o as well as the
Almkavist's estimation.
