Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I've been having a look at how Lieb and Loss (in their textbook on Analysis) prove Hanner's inequalities and have been trying to get a handle of the geometric intuition involved. In doing so I've been trying to prove that:

$(a+b)^p + (a-b)^p \geq 2a^p +p(p-1)a^{p-2}b^2$

For any $1\leq p\leq 2$ and $0<b<a$.

This is exercise 4 in chapter two of the aforementioned book. Can anyone give me any tips (or some geometric intuition)?

Also can anyone give me any explanation of why uniform convexity is important, and what it might mean geometrically?

Edit: for the latter question the following is useful but I would appreciate any more comments. http://mathoverflow.net/questions/27691/hanners-inequalities-the-intuition-behind-them

share|improve this question
add comment

1 Answer 1

Binomial Theorem.

$$(a+b)^p=a^p(1+(b/a))^p=a^p(1+(b/a)p+(b/a)^2{p\choose2}+\cdots)$$

$$(a-b)^p=a^p(1-(b/a))^p=a^p(1-(b/a)p+(b/a)^2{p\choose2}+\cdots)$$

Adding, $$(a+b)^p+(a-b)^p=a^p(2+2(b/a)^2{p\choose2}+\cdots)\ge2a^p+p(p-1)a^{p-2}b^2$$

share|improve this answer
    
Damn, that's way easier than everything I was trying. Thanks –  user63214 Feb 21 '13 at 0:23
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.