Hi. I have a question related to Turan`s problem, that is

## Find a sequence of polynomial $P_n(x)$ satisfying $P_{n+1}(x)P_{n-1}(x) < P_{n}^2(x)$.

I am considering the generalized question for positive multivarible, i.e.

Let $k$ be a positive integer and let $x_1, ... x_k$ be k indeterminates. For $x_1, ... x_n > 0$, find a sequence of polynomial $P_n(x_1, ... , x_k)$ satisfying

## $P_{n+1}(x_1, ... , x_k) P_{n-1}(x_1, ... , x_k) < A(n)P_n(x_1, ... , x_k)^2$ where

$A(n)$ is some fixed function for $n$.

Is there any result or some reference related to this problem?

In particular, I am interested in the case when $A(n) = \frac{n+2}{n+1}$. And I tried to check the above inequality by using maple by letting $P_n(x_1, ... , x_k) = x_1^n + ... + x_k^n$. then suprisingly(for me) I didn't find a counterexample until now, nor prove the inequality.

How can I prove (or find a counterexample) of this inequality? I really appriciate for your any comment and help.

Thank you in advance.