  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 $n\geq 1$, $a_n^2-a_{n-1}a_{n+1}\geq 0$. It is
said to satisfy the higher order Tur\'{a}n inequalities if for $n\geq 1$,
$4(a_{n}^2-a_{n-1}a_{n+1})(a_{n+1}^2-a_{n}a_{n+2})-(a_{n}a_{n+1}-a_{n-1}a_{n+2})^2\geq
0$. A sequence satisfying the Tur\'an inequalities is also called log-concave.
For the partition function $p(n)$, DeSalvo and Pak showed that for $n>25$, 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 $n\geq 95$. 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 $g_{3,n-1}(x)=p(n-1)+3p(n)x+3p(n+1)x^2+p(n+2)x^3$ have only real
zeros. We conjecture that for any positive integer $m\geq 4$ there exists an
integer $N(m)$ 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 independently
posed by Ono.
