I am sorry. Ottem is right. $P_n$ is a polynomial of degree $n$. In particular, you may assume that $P_n$ is a symmetric function. Let's focus that $P_n(x_1, ... ,x_k) = x_1^n + ... x_k^n$, for any $k > 0 , n > 1$ given integers.

By rearrangement inequality, we can easily show that $ F(n) = \frac{P_n^2}{P_{n+1}P_{n-1} \leq 1.$ I want to know about the lower bound of $F(n)$ for each $n$.

I am sorry. Ottem is right. $P_n$ is a polynomial of degree $n$. In particular, you may assume that $P_n$ is a symmetric function. Let's focus that $P_n(x_1,\, \dots ,x_k) = x_1^n + \cdots + x_k^n$, for any $k > 0, n > 1$ given integers.

By rearrangement of the inequality, we can easily show that $F(n) = \frac{P_n^2}{P_{n+1} P_{n-1}} \leq 1.$ I want to know about the lower bound of $F(n)$ for each $n$.