Inequality concerning nonnegative numbers (related to Hanner's inequalities)

Question

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

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