The Hardy's inequality for integrals asserts that
Let $f:(0,\infty) \rightarrow \mathbb R$ be in $L^p$, $1<p<\infty$. Let $F(x) = \frac{1}{x} \int_0 ^x f(t)dt$. Then $F\in L^p$ and $\|F\|_p \leq \frac{p}{p-1} \|f\|_p$.
My question is
When will the inequality becomes equality? How do we prove it?
As Wikipedia says, the equality holds if and only if $f(x) = 0$ almost everywhere. However, there is no proof for this on Wikipedia. It seems that the only inequality used in the proof of Hardy's inequality is Minkowski's inequality. But I can't obtain the desired conclusion from when Minkowski's inequality becomes an equality.
Any comments are appreciated.