Find the smallest constant C that satisfies the inequality: where z is any complex number.
I know that , but the answer is
Let z = x + iy
|Re(z)| = |x|
|Im(z)| = |y|
Expressed in polar form,
Substituting into the expression and simplifying we get
The greatest value of C occurs when (this can be proved using calculus - take the first derivative and find the maximum point).
so the greatest value C can be is
Therefore