# Trigonometric cancellation on the unit circle

Let $$z \in \mathbb{C}$$ with $$|z|=1$$ and $$z\ne 1$$. Now consider the sum $$S(N,p)=\sum_{k=0}^N k^p z^k,$$ for some positive integers $$N,p$$.

An immediate upper bound on $$|S(N,p)|$$ is $$|S(N,p)|\le C_1(p)N^{p+1},$$ for some constant $$C_1$$ depending only on $$p$$. I'm looking for a reference showing that accounting for the cancellation we have $$|S(N,p)|\le C_2(|1-z|,p)N^{p},$$ for some explicit constant $$C_2$$ depending only on $$p,|1-z|,$$ and monotonic decreasing in $$|1-z|$$.

It is possible to prove such a bound using $$\sum_{k=0}^n k^p z^k = \left(z \frac{d}{dz}\right)^p \frac{1-z^{N+1}}{1-z}$$, but since it's surly known I'm hoping for a reference.

We may simply write $$S(n,p)(1-z)=-N^pz^{N+1}+\sum_{k=1}^{N} z^k(k^{p}-(k-1)^p),$$ thus by triangle inequality $$\left|S(n,p)\right|\cdot |1-z|\leqslant 2N^p.$$