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Let $\mathcal{A}=(a_i)_{i=1}^\infty$ be a non-decreasing sequence of positiveintegers 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 inequalityfor the function $p_\mathcal{A}(n,k)$. Further, we discover when and under whatconditions on $k$, $\{a_1,a_2,\ldots,a_k\}$ and $N\in\mathbb{N}_+$, thesequence $\left(p_\mathcal{A}(n,k)\right)_{n=N}^\infty$ is log-concave. Ourproofs are based on the asymptotic behavior of $p_\mathcal{A}(n,k)$, inparticular, we apply the results of Netto and P\'olya-Szeg\"o as well as theAlmkavist's estimation.

Log-concavity of the restricted partition function $p_\mathcal{A}(n,k)$ and the new Bessenrodt-Ono type inequality