Question Number 149585 by mathdanisur last updated on 06/Aug/21

if x;y;z>0 and x^2 +y^2 +z^2 =3 then: Σ ((x^2 +y^2 )/((2x^2 +y^2 )(y^2 +2x^2 ))) ≥ (2/3)

Answered by dumitrel last updated on 06/Aug/21

(√((2x^2 +y^2 )(2y^2 +x^2 )))≤^(am−gm) ((3(x^2 +y^2 ))/2)⇒ (2x^2 +y^2 )(2y^2 +x^2 )≤((9(x^2 +y^2 )^2 )/4) Σ((x^2 +y^2 )/((2x^2 +y^2 )(2y^2 +x^2 )))≥Σ((x^2 +y^2 )/((9(x^2 +y^2 )^2 )/4))= (4/9)Σ(1/(x^2 +y^2 ))≥(4/9)∙(9/(2(x^2 +y^2 +z^2 )))=(2/3)

Commented bymathdanisur last updated on 06/Aug/21

Thank you Ser, cool