Could use some hint/help here. Do not really know where to start, showing the (in)equality. My guess would be some kind of generalized parallelogram inequality?

Let $0 < r \leq 1$ and $1 \leq p < \infty$, then define the following:

\begin{align*} \alpha\left( r \right) &= \left( 1 + r \right)^{p - 1} + \left( 1 - r \right)^{p-1} \\ \beta\left( r \right) &= r^{1 - p} \left( \left( 1+r \right)^{p-1} - \left( 1 - r \right)^{p-1} \right) \end{align*}

Then, it holds for $1 \leq p \leq 2$: \begin{equation*} \vert a \vert^p \alpha(r) + \vert b \vert^p\beta(r) \leq \vert a + b \vert^p + \vert a-b \vert^p \quad \forall a,b \in \mathbb{R} \end{equation*}

and for $2 \leq p < \infty$ (sic) equality holds.


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