Help proving complex inequality

[JerryG]

This may seem trivial, but for some reason I am having trouble with it. For a and b in the complex plane, I am trying to prove the following:

|a|^2+|b|^2 >= |(a+b)/2|^2

I need this for part of a larger proof.

 

[Guffel]

Since no one has answered yet, I'll give it a go.

Starting from the triangle inequality, we get

|a+b| <= |a| + |b|

=>

|a+b|^2 <= (|a| + |b|)^2 = |a|^2 + |b|^2 + 2|a||b|

=>

|(a+b)/2|^2 <= |a|^2 / 4 + |b|^2 / 4 + |a||b| / 2If we can prove that |a|^2 + |b|^2 >= |a|^2 / 4 + |b|^2 / 4 + |a||b| / 2, then we're done!

Can you go from here?