If z is a non-zero complex number, prove that
|conjugate(z) + 1/z| >=2
$\displaystyle |\bar{z}+1/z|\geq 2\Leftrightarrow |\bar{z}+1/z|^2\geq 4$ . Using $\displaystyle |w|^2=\bar{w}w$ we obtain
$\displaystyle |\bar{z}+1/z|^2=\ldots =|z|^2+2+1/|z|^2\geq \ldots \geq 4$ .