Several ways to prove Hardy's inequality

Suppose $1 < p < \infty$ and $f \in L^p((0,\infty))$ (with respect to Lebesgue measure of course) is a nonnegative function, takeF(x) = \frac{1}{x}\int_0^x f(t)dt \quad 0 < x <\infty,we have Hardy’s inequality $\def\lrVert[#1]{\lVert #1 \rVert}$\lrVert[F]_p \leq q\lrVert[f]_pwhere $\frac{1}{p}+\frac{1}{q}=1$ of course.There are several ways to prove it. I think there are several good reasons to write them down thoroughly since that may be why you find this page. Maybe you are burnt out since it’...

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