1. Upperbound (Complex Analysis)

I am trying to figure out the following problem:
Find an upper bound on |
z3-2z2+6z+2| defined by |z|<2.

I've done the following:
|z3-2z2+6z+2|
= |(x+iy)3-2(x+iy)2+6(x+iy)+2|
= |x3+3ix2y-3xy2-iy3-2(x2+2ixy-y2)+6x+6iy+2|
= |x3+3ix2y-3xy2-iy3-2x2-4ixy+2y2+6x+6iy+2|
= |x3-3xy2-2x2+2y2+6x+2+(3x2y-y3-4xy+6y)i|

= √(x3-3xy2-2x2+2y2+6x+2)2+ (3x2y-y3-4xy
+6y)2

I don't think I'm going this the right way. Any ideas? Should I not change to x+iy form?

2. Re: Upperbound (Complex Analysis)

by repeated applications of the triangle inequality

$\left|z^3-2z^2+6z+2\right| \leq |z^3| + |2z^2|+|6z|+|2| = |z|^3 + 2 |z|^2 + 6|z| + 2$

and if $|z|< 2$ then

$\left|z^3-2z^2+6z+2\right| < 2^3 + 2\cdot 2^2 + 6\cdot 2 + 2 = 30$

3. Re: Upperbound (Complex Analysis)

I'm sorry, guys. I read the problem wrong. I need to use the triangle inequalities. I overcomplicated this problem.

4. Re: Upperbound (Complex Analysis)

Originally Posted by romsek
by repeated applications of the triangle inequality

$\left|z^3-2z^2+6z+2\right| \leq |z^3| + |2z^2|+|6z|+|2| = |z|^3 + 2 |z|^2 + 6|z| + 2$

and if $|z|< 2$ then

$\left|z^3-2z^2+6z+2\right| < 2^3 + 2\cdot 2^2 + 6\cdot 2 + 2 = 30$
Thanks! I figured it out as soon as I finished typing it and submitting it. I over complicated things! Any way I can delete this post since it doesn't really contribute much to the community?