Question

Let
z1,z2,...ϵC
z1,z2,...ϵC
then the reverse triangle inequality is given by .......

Answer 1

Answer on Question #58293 – Math – Complex Analysis

Let $z_1, z_2 \in \mathbb{C}$. Then the reverse triangle inequality is given by ...

Solution:

The triangle inequality is given by

$$\left| z _ {1} + z _ {2} \right| \leq \left| z _ {1} \right| + \left| z _ {2} \right|$$

Then

$$\left| z _ {1} \right| = \left| z _ {1} + \left(z _ {2} - z _ {2}\right) \right| = \left| \left(z _ {1} - z _ {2}\right) + z _ {2} \right|$$

Using the triangle inequality we obtain:

$$\left| z _ {1} \right| \leq \left| \left(z _ {1} - z _ {2}\right) \right| + \left| z _ {2} \right|$$

So

$$\left| z _ {1} \right| - \left| z _ {2} \right| \leq \left| z _ {1} - z _ {2} \right|$$

As $|z| = |-z|$ we obtain (permutation $z_1 \rightleftarrows z_2$) the reverse triangle inequality:

$$\left| \left| z _ {1} \right| - \left| z _ {2} \right| \right| \leq \left| z _ {1} - z _ {2} \right|$$

**Answer**: The reverse triangle inequality is given by $\left| \left| z_1\right| - \left| z_2\right| \right| \leq \left| z_1 - z_2 \right|$.

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