Bounds for Weierstrass Elementary Factors

Theorem
Let $E_p:\C\to\C$ denote the $p$th Weierstrass elementary factor:


 * $E_p \left({z}\right) = \displaystyle \begin{cases} 1 - z & : p = 0 \\

\left({1 - z}\right) \exp \left({z + \dfrac {z^2} 2 + \cdots + \dfrac{z^p} p}\right) & : \text{otherwise}\end{cases}$

Let $z\in\C$.

Some bound
If $|z|\leq\frac12$, then $|E_p(z) - 1| \leq 3|z|^{p+1}$.

Another bound
If $|z|\leq 1$, then $|E_p(z) - 1| \leq |z|^{p+1}$.

Proof of some bound
Let $|z|\leq\frac12$.

We may assume $p\geq1$.

We have:
 * $E_p(z) = \exp\left( \log(1-z) + \sum_{k=1}^p\frac{z^k}k \right)$

and:

Because $p\geq1$, $2\vert z\vert^{p+1}\leq\frac12$.

By Bounds for Complex Exponential, $|E_p(z) - 1| \leq 3|z|^{p+1}$

Also see

 * Weierstrass Factorization Theorem, what this is made for
 * Bounds for Complex Exponential
 * Bounds for Complex Logarithm