The elementary symmetric functions of a reciprocal polynomial sequence

Erdös 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)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if n?4$n?4$, then none of the elementary symmetric functions of 1/m,1/(m+d),…,1/(m+(n−1)d)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ 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)$1/f(1),1/f(2),\dots ,1/f(n)$ is an integer except for f(x)=x^m$f(x)=x^m$ with m?2$m?2$ being an integer and n=1$n=1$.