I am having a look at the chapter "Uniform Convexity and Uniform Smoothness" from the book "Geometry and Martingales in Banach Spaces" by Woyczynski.
I would like to prove Example 3.1.2: For $p>1$, the $L^p$-space is $p\wedge 2$-uniformly smooth when equipped with the $L^p$-norm.
I think that I have found a proof, but I am not quite sure where I used the fact that $p$ may not be bigger than 2. What I have gathered is the follwoing: Let $f,g\in L^p$. Then we have: $$||(f+g)/2||+ ||(f-g)/2||-1 = \frac{1}{2} \left( \int |f(x)+g(x)|^p dx \right)^{1/p} +\frac{1}{2} \left( \int |f(x)-g(x)|^p dx \right)^{1/p} -1 \\ \overset{(1)}{\leq} \left( \int \frac{1}{2}|f(x)+g(x)|^p + \frac{1}{2}|f(x)-g(x)|^p dx \right)^{1/p} -1 \\ \overset{(2)}{\leq} \left( \int |f(x)|^p +|g(x)|^p dx \right)^{1/p} -1 \\ = \left( ||f||^p+||g||^p \right)^{1/p} -1\\ = (1+\tau^p)^{1/p} -1 \\ \leq \tau^p $$ where (1) uses that the $p$-th root is concave and (2) is just the fact that $|a+b|^p+|a-b|^p\leq |a|^p + |b|^p.$
I think my problem is less about uniform smoothness and more about the calculations that are done here. I suspect $|a+b|^p+|a-b|^p\leq |a|^p + |b|^p$ to hold, but find myself unable to prove this or find a source. Also, I cannot imagine how this would only hold for $p>2$.
I am thankful for any hints.