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. 


Answers are much appreciated. 



Trigonometric cancellation on the unit circle
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
$$
\left|S(n,p)\right|\cdot |1-z|\leqslant 2N^p.
$$


