Turán inequalities for the plane partition function

Abstract

Heim, Neuhauser and Tröger 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 Turán inequalities for sufficiently large n. The case where d=2 is the case of log-concavity.