If we consider $0<r<1$ and $j\in\mathbb{N}$, is this inequality true?
$$\left|\sum_{k=0}^jr^ke^{ikt}\right|\leq \left|\sum_{k=0}^je^{ikt}\right|$$
EDIT: A counterexample is in the comments. I'm wondering, however, if there is actually an inequality when you take integrals:
$$\int_0^{2\pi}\left|\sum_{k=0}^jr^ke^{ikt}\right|dt\leq \int_0^{2\pi}\left|\sum_{k=0}^je^{ikt}\right|dt$$