- #1

- 58

- 0

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

I need this for part of a larger proof.

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- Thread starter JerryG
- Start date

- #1

- 58

- 0

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

I need this for part of a larger proof.

- #2

- 28

- 0

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| / 2

If 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?

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