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:
$$\int_0^{2\pi}\left|\sum_{k=0}^jr^ke^{ikt}\right|dt\leq \int_0^{2\pi}\left|\sum_{k=0}^je^{ikt}\right|dt$$
Going to answer myself. The inequality with the integrals is true. If you call $g(t):=\sum_{k=0}^je^{ikt}\in P(\mathbb{T})\subset L^1(\mathbb{T})$ and denote the Poisson kernel by $P_r(t)=\sum_{k\in\mathbb{Z}}r^{|k|}e^{ikt}$, using Young's inequality for $p=1$ you have:
$$\int_0^{2\pi}\left|\sum_{k=0}^jr^ke^{ikt}\right|dt=\|P_r\ast g \|_1\leq \|P_r\|_1\cdot\|g\|_1= \|g\|_1=\int_0^{2\pi}\left|\sum_{k=0}^je^{ikt}\right|dt$$