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

Let $\mathcal{A}=(a_i{)}_{i=1}^{\mathrm{\infty }}$ 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 $\{a_1,a_2,\dots ,a_k\}$. 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 $k$, $\{a_1,a_2,\dots ,a_k\}$ and $N\in {\mathbb{N}}_{+}$, the sequence ${(p_{\mathcal{A}}(n,k))}_{n=N}^{\mathrm{\infty }}$ 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 Pólya-Szegö as well as the Almkavist’s estimation.