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The Tur\'{a}n inequalities and the higher order Tur\'{a}n inequalities arisein the study of Maclaurin coefficients of an entire function in theLaguerre-P\'{o}lya class. A real sequence $\{a_{n}\}$ is said to satisfy theTur\'{a}n inequalities if for $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It issaid to satisfy the higher order Tur\'{a}n inequalities if for $n\geq 1$,$4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq0$. A sequence satisfying the Tur\'an inequalities is also called log-concave.For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, thesequence $\{ p(n)\}_{n> 25}$ 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 orderTur\'{a}n inequalities for $n\geq 95$. In this paper, we prove this conjectureby using the Hardy-Ramanujan-Rademacher formula to derive an upper bound and alower bound for $p(n+1)p(n-1)/p(n)^2$. Consequently, for $n\geq 95$, the Jensenpolynomials $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only realzeros. We conjecture that for any positive integer $m\geq 4$ there exists aninteger $N(m)$ such that for $n\geq N(m) $, the polynomials $\sum_{k=0}^m{m\choose k}p(n+k)x^k$ have only real zeros. This conjecture was independentlyposed by Ono.

Higher Order Tur\'an Inequalities for the Partition Function