Is there $C > 0$ such that the inequality
$$
\prod\_{n\in\mathbb{N}} p(n)^{a\_n} \leq p\left(C\prod\_{n\in\mathbb{N}} n^{a\_n}\right)
$$
holds for all finitely supported sequences $(a\_n)$ with $a\_n\geq 0$ and $\sum\_n a\_n = 1$ and polynomials with nonnegative coefficients $p\in \mathbb{R}\_+[X]$?


Is it even possible to take $C=1$?


It's clear that the inequality trivially holds with $C = 1$ whenever $p$ is a monomial.



A kind of exponential concavity for polynomials?
The example by Philipp Lampe can be modified to prove that no such $C$ exists:


Suppose $k$ is an arbitrary positive integer and take $p(x)=x(x+k^2-1)$. Now choose $a\_1=a\_n=\frac{1}{2}$ for some $n\geq 1$ and we get that $C$ must satisfy
$$\sqrt{p(1)p(n)}\le p(C\sqrt{n})$$
$$\iff \sqrt{k^2n(n+k^2-1)}\le C\sqrt{n}(C\sqrt{n}+k^2-1)$$
but we notice that as we let $n\to \infty$ the left side is $\sim kn$ and the right side is $\sim C^2 n$ which gives $C\geq \sqrt{k}$. Since $k$ was arbitrary we conclude that no such constant exists.


