Let $a, b, c, d \in \mathbb{R}$ and $a, b, c, d\gt 1$.
For any polynomial $A_n = \sum_{i=0}^n a_ix^i$, i'm interested in a quantity $g(A_n) = \frac{1}{n}\langle (a_0, \cdots, a_n), (1/{n \choose 0}, \cdots, 1/{n \choose n})\rangle$. Here $\langle,\rangle$ is the inner product.
Now my question is the relationship between $g((1+ax)(1+bx)(1+cdx))$ and $g((1+ax)(1+bx)(1+cx)(1+dx))$ ?
Is the first term always larger than the second one?
and does this pattern hold for arbitrary degrees of polynomial?
e.g. $\forall a_i, b_i \gt 1, n\ge 3, m\ge2$
would this always hold $$ g(A_n(1+\prod_{i=0}^mb_ix)) \gt g(A_n\prod_{i=0}^m (1+b_ix) $$ ?