Capacity Lower Bounds via Productization

The purpose of this note is to state and prove a lower bound on the capacity of a real stable polynomial $p(x)$ which is based only on its value and gradient at $x = 1$. This can be seen as a quantitative generalization of the fact that the capacity of $p$ is maximized when the normalized gradient of $p$ at 1 is equal to 1. This result implies a sharp improvement to a similar inequality proved by Linial-Samorodnitsky-Wigderson in 2000. Such inequalities have played an important role in the recent work on operator scaling and its generalizations and applications.