We are asked to show the uniform convexity of $L^p$ for $1 < p \le 2$ using the following inequality:

For all $1 < p < \infty$, there is a constant $C$ such that $|a - b|^p \leq C(|a|^p + |b|^p)^{1-s}\big(|a|^p + |b|^p - 2|\frac{a + b}{2}|^p\big)^s$ where $s = p/2$ for all $a, b \in \mathbb{R}$

I have actually managed to prove this inequality and I think reaching uniform convexity from this step should be easy but I'm having trouble with this final step!

I think we're supposed to use this inequality above with Holder's inequality but I'm unsure how exactly.



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