Higher Order Turan Inequalities for the Partition Function

The Tur{a}n inequalities and the higher order Tur{a}n inequalities arise in the study of Maclaurin coefficients of an entire function in the Laguerre-P{o}lya class. A real sequence ${a_{n}}$ is said to satisfy the Tur{a}n inequalities if for $ngeq 1$, $a_n^2-a_{n-1}a_{n+1}geq 0$. It is said to satisfy the higher order Tur{a}n inequalities if for $ngeq 1$, $4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2geq 0$. A sequence satisfying the Turan inequalities is also called log-concave. 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 Chen that $p(n)$ satisfies the higher order Tur{a}n inequalities for $ngeq 95$. 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 $ngeq 95$, the Jensen polynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real zeros. We conjecture that for any positive integer $mgeq 4$ there exists an integer $N(m)$ such that for $ngeq N(m) $, the polynomials $sum_{k=0}^m {mchoose k}p(n+k)x^k$ have only real zeros. This conjecture was independently posed by Ono.