Higher Turán inequalities for the plane partition function

Here we study the roots of the doubly infinite family of Jensen polynomials \(J_{\textrm{PL}}^{d,n}(x)\) $J_{\text{PL}}^{d,n}(x)$ associated to MacMahon’s plane partition function \(\textrm{PL}(n)\) $\text{PL}(n)$ . Recently, Ono et al. [Ono et al. in Adv Math 409:108692, 2022] proved that \(\textrm{PL}(n)\) $\text{PL}(n)$ is log-concave for all \(n\ge 12\) $n\ge 12$ , which is equivalent to the polynomials \(J_{\textrm{PL}}^{2,n}(x)\) $J_{\text{PL}}^{2,n}(x)$ having real roots. Moreover, they proved, for each \(d\ge 2\) $d\ge 2$ , that the \(J_{\textrm{PL}}^{d,n}(x)\) $J_{\text{PL}}^{d,n}(x)$ have all real roots for sufficiently large n. Here we make their result effective. Namely, if \(N_{\textrm{PL}}(d)\) $N_{\text{PL}}(d)$ is the minimal integer such that \(J_{\textrm{PL}}^{d,n}(x)\) $J_{\text{PL}}^{d,n}(x)$ has all real roots for all \(n\ge N_{\textrm{PL}}(d)\) $n\ge N_{\text{PL}}(d)$ , then we show that \(\begin{aligned} N_{\textrm{PL}}(d)\le 279928\times d(d-1)\, \left( 6 d^3\, (22.2)^{\frac{3(d-1)}{2}}\right) ^{2d} e^{\frac{\Gamma (2d^2)}{(2\pi )^{2d+2}}}. \end{aligned}\) $$\begin{array}{c}\hfill N_{\text{PL}}(d)\le 279928\times d(d-1){\left(6,d^3,,{(22.2)}^{\frac{3(d-1)}{2}}\right)}^{2d}{e}^{\frac{\mathrm{\Gamma }(2d^2)}{{(2\pi )}^{2d+2}}}.\end{array}$$ Moreover, using the ideas that led to the above inequality, we explicitly prove that \(N_{\textrm{PL}}(3)=26, N_{\textrm{PL}}(4)=46, N_{\textrm{PL}}(5)=73, N_{\textrm{PL}}(6)=102\) $N_{\text{PL}}(3)=26,N_{\text{PL}}(4)=46,N_{\text{PL}}(5)=73,N_{\text{PL}}(6)=102$ and \(N_{\textrm{PL}}(7)=136\) $N_{\text{PL}}(7)=136$ .