Tur?n inequalities for the plane partition function

Heim, Neuhauser and Troger recently established some inequalities for MacMahon's plane partition function PL(n) that generalize known results for Euler's partition function p(n). They also conjectured that PL(n) is log-concave for all n > 12. We prove this conjecture. Moreover, for every d > 1, we prove their speculation that PL(n) satisfies the degree d Turan inequalities for sufficiently large n. The case where d = 2 is the case of log-concavity.(c) 2022 Elsevier Inc. All rights reserved.