Stronger lower bound for geometric sums

Prove the stronger inequality \(|1+z+\cdots+z^{n-1}|\geq\min(n,2\pi)\) for every positive integer \(n\) and every complex number \(z\) in the closed disk \(\{|z-2|\leq 1\}\).

Background

A central technical component of the paper is a lower bound for the modulus of the geometric sum $(z)_n=1+z+\cdots+z^{n-1}$ on the disk centered at two with radius one. The paper establishes the weaker estimate $|(z)_n|\geq\min(n,11/2)$, which is sufficient for the main zero-exclusion theorem.

The authors identify $\min(n,2\pi)$ as a conjecturally sharp stronger bound. They explicitly note that their methods do not presently extend to this inequality without extensive computer assistance.