Question

a) Fie z ∈ C.Aratati ca daca $2z+3\bar{z} \in R$ , atunci z ∈ R.


b) Fie z ∈ C.Aratati ca daca $z^2 + \bar{z}^2 \ \geq \ 2|z|^2$ ,atunci z ∈ R.

Answer 1

a)z= a+bi; a, b∈RAvem2(a+bi)+3(a-bi) ∈ R2a+2bi+3a-3bi ∈R5a -bi ∈R ⇒ b=0 ⇒ z=a ∈Rb)$(a+bi)^2+(a-bi)^2 \geq 2 \sqrt{a^2+b^2}^2 \\ a^2+2abi-b^2+a^2-2abi-b^2 \geq 2a^2+2b^2 \\ 2a^2-2b^2 \geq 2a^2+2b^2 \\ -b^2 \geq b^2 \Leftrightarrow b=0 \Rightarrow z=a\in R$