Solving Inequality Problem: (|a|+|b|≥2(|c|+|d|))

[drago]

Hi,
I would appreciate some ideas on the following problem:

We are given the inequality:
(1)
|a| + |b| >= 2(|c| + |d|)

Can we conclude that:
(2)
(c + a)^2 + (d + b)^2 >= c^2 + d^2 ?

a, b, c, d are real.

I can prove (2) if (1) is in the form: |a| + |b| >= 4max(|c|,|d|), but the above is stronger.

Thank you.
drago

 

[TenaliRaman]

i feel its true
its seems so easy but at 12.30 midnight i cannot think of anything!

-- AI

 

[uart]

i feel its true
its seems so easy but at 12.30 midnight i cannot think of anything!

Yep I looked at it for a few minutes late last night and thought it looked easy but couldn't come up with much. I got another weaker result pretty easy but then decided to leave it for someone else.

BTW, I got the weaker result by assumming (without loss of generality) that |b| >= |a|. I then found that if I made the additional imposition that |c|>=|d| then I was able to prove the desired result pretty easily . This however definitely does give a loss of generality and hence a weaker result.