Turán inequalities for the broken k-diamond partition functions

We obtain an asymptotic formula for Andrews and Paule’s broken k -diamond partition function Δ k ( n ) , where k = 1 or 2. Based on this asymptotic formula, we derive that Δ k ( n ) satisfies the order d Turán inequalities for d ≥ 1 and for sufficiently large n when k = 1 or 2 by using a general result of Griffin, Ono, Rolen, and Zagier. We also show that Andrews and Paule’s broken k -diamond partition function Δ k ( n ) is log-concave for n ≥ 1 when k = 1 or 2, which implies that Δ k ( a ) Δ k ( b ) ≥ Δ k ( a + b ) for a , b ≥ 1 when k = 1 or 2.