the signe of a term

Question

let $\alpha, \beta \in \mathbb{R}$ two real numbers such that $\alpha<0$ and $\beta >0$ and $\alpha+\beta >0 $

and let be $Z\ \in \mathbb{C}$ and $|Z|$ the complex modulus of $Z$

what can i said about the sign of $\alpha |Z|^2+\beta |Z^2|$ ?

Answer 1

$\alpha |Z|^2+\beta |Z^2| = \alpha |Z|^2+\beta |Z|^2 = (\alpha+\beta) |Z|^2$ is positive and strictly positive provided $Z\not=0$.