Log-concavity of the restricted partition function [formula omitted] and the new Bessenrodt-Ono type inequality

Abstract

Let A=(ai)i=1∞ be a non-decreasing sequence of positive integers and let k∈N+ be fixed. The function pA(n,k) counts the number of partitions of n with parts in the multiset {a1,a2,…,ak}. We find out a new Bessenrodt-Ono type inequality for the function pA(n,k). Further, we discover when and under what conditions on k, {a1,a2,…,ak} and N∈N+, the sequence (pA(n,k))n=N∞ is log-concave. Our proofs are based on the asymptotic behavior of pA(n,k) — in particular, we apply the results of Netto and Pólya-Szegő as well as the Almkavist's estimation.