Given a polynomial with complex coefficients $p(x)=a_n x^n+\dots+a_1 x+a_0\in\mathbb{C}[x]$, I am trying to estimate the sum
$$ \sum_{i=0}^n |a_i|.$$
I was wondering if maybe this sum could be an integral on a specific contour, or have some relation to the maximum of $|p(x)|$ on the unit circle, or something like that.
Another approach that would be useful is: if I have a factorization of $p(x)$, say $p(x)=p_1(x)^{r_1}\dots p_m(x)^{r_m}$ -that is, if I start from the $p_i(x)$ and construct $p(x)$ as this product-, would there be a way (a simpler way than using multinomial expansions, ideally) of bounding $\sum |a_i|$ in terms of the coefficients of the $p_i(x)$?