  Erd\"{o}s and Niven proved in 1946 that for any positive integers $m$ and
$d$, there are at most finitely many integers $n$ for which at least one of the
elementary symmetric functions of $1/m, 1/(m+d), ..., 1/(m+(n-1)d)$ are
integers. Recently, Wang and Hong refined this result by showing that if $n\geq
4$, then none of the elementary symmetric functions of $1/m, 1/(m+d), ...,
1/(m+(n-1)d)$ is an integer for any positive integers $m$ and $d$. Let $f$ be a
polynomial of degree at least $2$ and of nonnegative integer coefficients. In
this paper, we show that none of the elementary symmetric functions of $1/f(1),
1/f(2), ..., 1/f(n)$ is an integer except for $f(x)=x^{m}$ with $m\geq2$ being
an integer and $n=1$.
