$\|f+g\|^p +\|f-g\|^p \geq (\|f\| +\|g\|)^p + | \|f\| -\|g\||^p$.

Question

Suppose $f, g\in L^p, 1<p<\infty$, is it true that $\|f+g\|^p +\|f-g\|^p \geq (\|f\| +\|g\|)^p + \left| \|f\| -\|g\|\right|^p$?

Here $\|f\| = (\int |f|^p)^{1/p}$ is $L_p$ norm.

Answer 1

For $\infty>p>2$ the inequality $$(\|f\|+\|g\|)^p+|\|f\|-\|g\||^p \geq \|f+g\|^p + \|f-g\|^p$$ holds and for $1 <p <2$ the reverse inequality $$(\|f\|+\|g\|)^p+|\|f\|-\|g\||^p \leq \|f+g\|^p + \|f-g\|^p$$ holds. That is Theorem 1 in Hanner's proof of the uniform convexity of the $L^p[0,1]$-spaces. Moreover, there is also a condition for equality: Equality holds if and only if $$(f(t)+ag(t))(f(t)-ag(t))=0$$ for almost every $t \in [0,1]$ for some $a \in \mathbb{R}$.

The inequalites also hold for arbitary measure spaces $(\Omega,\Sigma,\mu)$, provided $\mu$ is a positive measure. A proof can be found in here. In this paper you can also find an interesting extension of Jensen's inequality.