# Brezis excercise 4.12: $L_p$ is uniformly convex for $1<p\leq 2$

1. Let $$1. Prove that there is a constant $$C$$(depending only on $$p$$) such that $$|a-b|^p\leq C(|a|^p+|b|^p)^{1-s}(|a|^p+|b|^p-2|\dfrac{a+b}{2}|^p)^s$$ for all $$a,b\in \mathbb{R}$$ and $$s=\dfrac{p}{2}$$.

2.Deduce that $$L_p$$ is uniformly convex for $$1.

I have an idea on how to prove 2, assuming 1 and using Holder's inequality I think I can do it, however I dont see a way to prove 1. any hints? Thanks in advance.

• This does not look homogenous. I would say $1-s$ instead of $s-1$. – Mindlack Jan 10 at 18:44
• Right. edited, thanks. – Alfdav Jan 10 at 18:48
• You may assume wlog that $a = 1$, due to homogeneity. Then it's essentially a one variable calculus problem. – Hans Engler Jan 10 at 20:30