How do you obtain the inequality $ |1+e^{2(R+iy)}| \geq |e^{2(R+iy)| -1$?

Question

How do you get that $|1+e^{2(R+iy)}| \geq |e^{2(R+iy)}| -1$?

I know that this is sort of the reverse triangle inequality at work, but I am not too sure how the lecturer does it.

We previously obtained that $$|1+e^{2(-R+iy)}| \geq 1 - |e^{2(-R+iy)}| $$ which I understand.

But here he has reversed the order of the 1 and the exponential and I don't understand why.

Answer 1

This follows directly from the inequality

$$|a-b|\geq ||a|-|b||$$

It means that both these are true:

$$|a-b|\geq|a|-|b|$$ $$|a-b|\geq|b|-|a|$$

Take $a=1$ and $b=-e^{2(R+iy)}$ or $b=-e^{2(-R+iy)}$, and conclude.

Answer 2

It's generally true that $|1+z|\ge |z|-1$. And yes, this is essentially the triangle inequality: $|z|=|1+z-1|\le |1+z|+|-1|=1+|1+z|$