$­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$?

Question

Is there a name for this formula? For $f_k,w_k\geqslant0$.

$$­\prod_{k=1}^{n}f_k\leqslant\sum_{q=1}^{n}w_q\prod_{p=1,p\neq q}^{n}f_p.$$

I believe that there is $w_k$ that make the formula true. Am I right?

EDIT: $f_k=(x_k-a)(x_k-b)$ and $w_k=x_k^2$, $a$ and $b$ are the roots of $f_k$.

So I want to prove this:

$$­\prod_{k=1}^{n}(x_k-a)(x_k-b)\leqslant\sum_{q=1}^{n}x_q^2\prod_{p=1,p\neq q}^{n}(x_p-a)(x_p-b).$$

Answer 1

In general that inequality does not hold.

Taking $x_1=a$ and $x_2=\cdots=x_{n-1}=0$ for example, the inequality becomes $(ab)^{n-2}(x_n-a)(x_n-b) \ge 0$, which is clearly not true in general. If $a\neq b$ then the sign of $(x_n-a)(x_n-b)$ can change (from $-\infty$ to $+\infty$, the order is $+-+$).