Complex Inequalities

Complex Inequalities

## Formula

$$|z|\ge|\operatorname{Re}(z)|$$

$$|z|\ge|\operatorname{Im}(z)|$$

For:

$$z=a+ib$$

$$|a|\le\sqrt{a^2+b^2}$$

$$|b|\le\sqrt{a^2+b^2}$$

## Conditions / Special Cases

Equality in:

$$|z|\ge|\operatorname{Re}(z)|$$

occurs when:

$$\operatorname{Im}(z)=0$$

Equality in:

$$|z|\ge|\operatorname{Im}(z)|$$

occurs when:

$$\operatorname{Re}(z)=0$$

## Usage

- Used to establish bounds involving real and imaginary parts.