2
$\begingroup$

Find all polynomials with real coefficients such that $$xp(x) + yp(y)\geq2p(xy)$$ for all $x, y\in\mathbb{R}$.

My attempt:

I noticed the similarity between the given expression and the square of the sum $$(x-y)^2 = x^2 + y^2 - 2xy\geq0$$ So I guessed that $p$ would be of the form $$p(x)=ax$$ and I found that $a>0$. I tried to do $p(x)=ax\cdot q(x) + r$ to proof that $p(x)=ax$ would be the only solution, but I didn't finish this.

Any help would be appreciated!

Thanks for attention.

$\endgroup$
1

0

You must log in to answer this question.

Browse other questions tagged .