Turán inequalities for the plane partition function (Q2089663)

Very recently \textit{B. Heim} et al. [``Inequalities for plane partitions'', Ann. Comb. (to appear)] have undertaken a study of MacMahon plane partition function \(PL(n)\) in analogy with some known results on Euler's partition function \(p(n)\). They prove that \(PL(n)\) is log-concave for sufficiently large \(n\), and they pose the following explicit conjecture: The function \(PL(n)\) is log-concave for \(n\geqslant 12\). The condition of log-concavity for nonvanishing real sequences \(\{\alpha_n\}\) is the first example of the Turán inequalities, which can be conveniently formulated in terms of Jensen polynomials of degree \(d\) and shift \(n\) \[J_{\alpha}^{d,n}(X)=\sum_{j=0}^d \binom{d}{j}\alpha(n+j) X^j.\] Related to Euler's partition function \(p(n)\), \textit{W. Y. C. Chen} et al. [Trans. Am. Math. Soc. 372, No. 3, 2143--2165 (2019; Zbl 1415.05020)] proved that \(J^{3,n}_p(X)\) is hyperbolic for \(n \geqslant 94\), and they pose the following conjecture: for every integer \(d\geqslant 1\), that there is an integer \(N_p(d)\) for which \(J^{d,n}_p(X)\) is hyperbolic for \(n\geqslant N_p(d)\). In this paper, the authors prove these conjectures. In addition, they prove that \(J^{d,n}_{PL}(X)\) is hyperbolic for all sufficiently large \(n\). The proof of these results is based on a strong asymptotic formula for \(PL(N)\) which was introduced by the authors in the paper.