- Let $1<p<\infty$. 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 $1<p\leq 2$.
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.