Bounds for Complex Logarithm

Theorem
Let $\log$ denote the complex logarithm.

Let $z\in\C$ with $|z|\leq\frac12$.

Then $\frac12|z| \leq |\log(1+z)| \leq \frac32|z|$.

Proof
By definition of complex logarithm:
 * $-\log(1+z) = \displaystyle \sum_{n\mathop=1}^\infty\frac{(-z)^n}n$

Thus

By the Triangle Inequality:
 * $\displaystyle \frac12 \leq \left\vert \frac{\log(1+z)}z \right\vert \leq \frac32$