I want to prove $|w\overline{z}+\overline{w}z|\leq 2|wz|$.
My attempt:
$\begin{array}{c c}|w\overline{z}+\overline{w}z| & =|(c+id)(a-ib)+(c-id)(a+ib)| \\ & =|2(ac+bd)| \\ & =\sqrt{4(ac+bd)^2} \\ & =2\sqrt{(ac+bd)^{2}}\end{array}$
Now
$\begin{array}{c c} 2|wz| & =2|(a+bi)(c+id)| \\ & =|2(ac-bd)+2i(ad+bc)| \\ & =2\sqrt{(ac-bd)^{2}+(ad+bc)^{2}}, \end{array}$
but $(ac+bd)^{2}>0$ y $(ac-bd)^{2}\geq 0$ so the second term is larger.
Is my procedure right?