# complex variables

• Nov 12th 2009, 09:37 AM
Gilbert
complex variables
Hi,

I need two proofs for complex inequalities.

1) |1-z|/z <2 or equal.

2) |1+z|/|z|>1/2 or equal

thanks
• Nov 12th 2009, 10:06 AM
tonio
Quote:

Originally Posted by Gilbert
Hi,

I need two proofs for complex inequalities.

1) |1-z|/z <2 or equal.

2) |1+z|/|z|>1/2 or equal

thanks

Proofs...of what? I think you have to describe the set of points on the (complex) plane that fulfill the given inequalities, so put $\displaystyle z=x+iy$ and (I think an absolute value (modulus) is lacking in the denominator of the first one):

$\displaystyle \frac{|1-z|}{|z|}\leq 2\Longrightarrow \frac{|(1-x)-yi|^2}{|x+iy|^2}\leq 4\Longrightarrow \frac{1-2x+x^2+y^2}{x^2+y^2}\leq 4$ $\displaystyle \Longrightarrow 3x^2 +3y^2+2x-1\geq 0\Longrightarrow 3\left(x+\frac{1}{3}\right)^2-\frac{1}{3}+3y^2-1\geq 0$ $\displaystyle \Longrightarrow \left(x+\frac{1}{3}\right)^2+y^2\geq \frac{4}{9}$

And we got a nice circle with center__ and radius__ . Now you try the second one.

Tonio
• Nov 12th 2009, 11:09 AM
Gilbert
complex variables
Hi,

maybe not a proof,but I have to show these inequalities.I need them for

the bound of a complex integral.

I taught something like that

|z+1|>|z-1|=1/2|z|+|1/2|z|-1|>1/2|z| but I'm not sure it's correct.

Tanks
• Nov 12th 2009, 11:21 AM
tonio
Quote:

Originally Posted by Gilbert
Hi,

maybe not a proof,but I have to show these inequalities.I need them for

the bound of a complex integral.

I taught something like that

|z+1|>|z-1|=1/2|z|+|1/2|z|-1|>1/2|z| but I'm not sure it's correct.

Tanks

You cannot "show those inequalities" because they're not true for all complex numbers z...for example, check that inequality (1) is false when $\displaystyle z=\frac{1}{2}i$ , and inequality (2) is false for example for $\displaystyle z=-1$

Tonio
• Nov 12th 2009, 11:56 AM
Gilbert
Complex variables
Hi,

yes you are right,but it's my fault.I have to show this inequalities for

z->infinity
as an upper bound and lower bound.

Sorry

Thanks