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

Abstract: 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.