Find the smallest constant C that satisfies the inequality: $\displaystyle |Re(z)|+|Im(z)|\leq C|z|$ where z is any complex number.
I know that $\displaystyle |Re(z)|+|Im(z)|\leq 2|z|$, but the answer is $\displaystyle C = \sqrt{2}$
Find the smallest constant C that satisfies the inequality: $\displaystyle |Re(z)|+|Im(z)|\leq C|z|$ where z is any complex number.
I know that $\displaystyle |Re(z)|+|Im(z)|\leq 2|z|$, but the answer is $\displaystyle C = \sqrt{2}$
Let z = x + iy
|Re(z)| = |x|
|Im(z)| = |y|
Expressed in polar form, $\displaystyle x = r\cos \theta ; y = r\sin \theta$
Substituting into the expression and simplifying we get
$\displaystyle r\cos\theta + r\sin\theta \leq Cr $
$\displaystyle \cos\theta + \sin\theta \leq C $
The greatest value of C occurs when $\displaystyle \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 $\displaystyle \cos \frac{\pi}{4} + \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt 2$
Therefore $\displaystyle C = \sqrt {2} $