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