1. 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.

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

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