Various proofs of Hardy's inequality For any $p > 1$ and for any sequence $\{a_j\}_{j=1}^\infty$ of nonnegative numbers, a classical inequality
of Hardy states that
$$ \sum\limits_{k=1}^n\left(\frac{\sum_{i=1}^ka_i}{k}\right)^p\le \left(\frac{p}{p-1}\right)^p \sum\limits_{k=1}^n a_k^p$$ 
for each $n\in  N$.
There are now many many proofs of Hardy's inequality. Which proof is your favourite one, which would be the simplest proof? It is preferable if you could present the detailed proof here so that everyone can share it. 
 A: Let $(\mathbb{R}^{+},\frac{dt}{t})$ be the multiplicative group of positive real numbers with the usual topology and Haar measure $\frac{dt}{t}$. Define functions $g:\mathbb{R}^{+}\to [0,\infty)$, $h:\mathbb{R}^{+}\to [0,\infty)$ by $g(x)=\left|f(x)\right|x^{1-\frac{b}{p}}$ and $h(x)=x^{-\frac{b}{p}}\chi_{[1,\infty)}(x)$. We will apply Minkowski's inequality to the convolution $F=g\star h$. Note that:
\begin{align*}
 F(x)= &\int_{0}^{\infty} \left|f(t)\right|t^{1-\frac{b}{p}}\;\;\frac{t^{\frac{b}{p}}}{x^{\frac{b}{p}}}\;\;\chi_{[1,\infty)}\left(\frac{x}{t}\right)\frac{dt}{t} \
     =& \frac{1}{x^{\frac{b}{p}}}\int_{0}^{x} \left|f(t)\right| dt \
\end{align*}
if $x\in\mathbb{R}^{+}$. Furthermore, 
$$
\left\|h\right\|_{L^1(\mathbb{R}^{+},\frac{dt}{t})}= \int_{1}^{\infty} t^{-\frac{b}{p}-1}dt=\frac{p}{b}
$$
and
$$\left\|g\right\|_{L^p(\mathbb{R}^{+},\frac{dt}{t})}=\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p-b-1} dt \right)^{\frac{1}{p}}
$$
Minkowski's inequality thus implies that 
\begin{align*}
\left(\int_{0}^{\infty} \left(\int_{0}^{x} \left|f(t)\right| dt\right)^p x^{-b-1} dx\right)^{\frac{1}{p}}\leq \frac{p}{b}\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p-b-1} dt\right)^{\frac{1}{p}}
\end{align*}
Let us now redefine the functions $g:\mathbb{R}^{+}\to [0,\infty), h:\mathbb{R}^{+}\to [0,\infty)$ by $g(x)=\left|f(x)\right|x^{1+\frac{b}{p}}$ and $h(x)=x^{\frac{b}{p}}\chi_{(0,1]}(x)$. We will apply Minkowski's inequality to the convolution $F=g\star h$. Note that, 
$$
F(x)= \int_{0}^{\infty} \left|f(t)\right|t^{1+\frac{b}{p}}\;\;\frac{x^{\frac{b}{p}}}{t^{\frac{b}{p}}}\chi_{(0,1]}\left(\frac{x}{t}\right)\frac{dt}{t}
     = x^{\frac{b}{p}}\int_{x}^{\infty} \left|f(t)\right| dt
$$
if $x\in\mathbb{R}^{+}$. Furthermore, 
$$
\left\|h\right\|_{L^1(\mathbb{R}^{+},\frac{dt}{t})}=\int_{0}^{1} t^{\frac{b}{p}-1} dt
                                                    = \frac{p}{b}
$$
and
$$\left\|g\right\|_{L^p(\mathbb{R}^{+},\frac{dt}{t})}=\left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p+b-1} dt \right)^{\frac{1}{p}}
$$
Minkowski's inequality thus implies that, 
\begin{align*}
\left(\int_{0}^{\infty} \left(\int_{x}^{\infty} \left|f(t)\right| dt\right)^p x^{b-1} dx\right)^{\frac{1}{p}}\leq \frac{p}{b} \left(\int_{0}^{\infty} \left|f(t)\right|^p t^{p+b-1} dt\right)^{\frac{1}{p}}
\end{align*}
A: I'd like to share this article. HARDY TYPE INEQUALITIES VIA CONVEXITY - THE JOURNEY SO FAR
http://ajmaa.org/searchroot/files/pdf/v7n2/v7i2p18.pdf 
It is refreshing!!!
