Inequalities for the broken k-diamond partition functions

In 2007, Andrews and Paule introduced the broken k -diamond partition function Delta k(n). Many researches on the arithmetic properties for Delta k(n) have been done. In this paper, we prove that D3 log Delta 1(n - 1) > 0 for n > 5 and D3 log Delta 2(n - 1) > 0 for n > 7, where D is the difference operator with respect to n. We also conjecture that for any k > 1 and r > 1, there exists a positive integer nk(r) such that for n > nk(r), (-1)rDr log Delta k(n) > 0. This is analogous to the positivity of finite differences of the logarithm of the partition function, which has been proved by Chen, Wang, and Xie. Furthermore, we obtain that both {Delta 1(n)}n >= 0 and {Delta 2(n)}n >= 0 satisfy the higher order Turan inequalities for n > 6.(c) 2023 Elsevier Inc. All rights reserved.