Thread: Find the smallest constant that satisfies the inequality

1. Find the smallest constant that satisfies the inequality

Find the smallest constant C that satisfies the inequality: $|Re(z)|+|Im(z)|\leq C|z|$ where z is any complex number.

I know that $|Re(z)|+|Im(z)|\leq 2|z|$, but the answer is $C = \sqrt{2}$

2. $z = |z|e^{i\theta} = |z|cos(\theta) + i|z|\sin(\theta)$ ?

3. Originally Posted by Mondreus
Find the smallest constant C that satisfies the inequality: $|Re(z)|+|Im(z)|\leq C|z|$ where z is any complex number.

I know that $|Re(z)|+|Im(z)|\leq 2|z|$, but the answer is $C = \sqrt{2}$
Let z = x + iy
|Re(z)| = |x|
|Im(z)| = |y|

Expressed in polar form, $x = r\cos \theta ; y = r\sin \theta$

Substituting into the expression and simplifying we get
$r\cos\theta + r\sin\theta \leq Cr$

$\cos\theta + \sin\theta \leq C$

The greatest value of C occurs when $\theta = \frac{\pi}{4}$ (this can be proved using calculus - take the first derivative and find the maximum point).

so the greatest value C can be is $\cos \frac{\pi}{4} + \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt 2$

Therefore $C = \sqrt {2}$