I am not sure how  to answer my own question on the researchgate website.

The solution is really trivial.

My question: how to show that 2*( x^2+y^2) >= (x+y)^2

My solution: 

2*(x^2+y^2) >= (x+y)^2

2x^2+2y^2 >= x^2+y^2+2xy

x^2+y^2-2xy >=0

(x-y)^2>=0 yes always

Cheers! MMarion