Higher order Turán inequalities for the partition function

Abstract:

The Turán inequalities and the higher order Turán inequalities arise in the study of the Maclaurin coefficients of real entire functions in the Laguerre–Pólya class. A sequence $\{a_{n}\}_{n\geq 0}$ of real numbers is said to satisfy the Turán inequalities or to be log-concave if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is said to satisfy the higher order Turán inequalities if for $n\geq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq 0$. For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, the sequence $\{ p(n)\}_{n> 25}$ is log-concave, that is, $p(n)^2-p(n-1)p(n+1)>0$ for $n> 25$. It was conjectured by the first author that $p(n)$ satisfies the higher order Turán inequalities for $n\geq 95$. In this paper, we prove this conjecture by using the Hardy–Ramanujan–Rademacher formula to derive an upper bound and a lower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\geq 95$, the Jensen polynomials $p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only distinct real zeros. We conjecture that for any positive integer $m\geq 4$ there exists an integer $N(m)$ such that for $n\geq N(m)$, the Jensen polynomial associated with the sequence $(p(n),p(n+1),\ldots ,p(n+m))$ has only real zeros. This conjecture was posed independently by Ono.