It is easy to see graphically that:
$$ |\eta(1.5 + ti) - \eta(-0.5+ti)| \ge |\eta(0.5+ti)| \forall t > 0, t \in \mathbb{R}$$
where $\eta(s)$ is defined as the analytical continuation of Dirichlet $\eta$.
I ran out of ideas on how to formally prove this. Any hint?