Proving Inequality: Expert Assistance for Your Inequality Proof

[Aaron792]

Can anyone help me prove this inequality?
See upload~

 

[Aaron792]

A complete description
Prove inequality
$\begin{equation} \int^\infty_0y^\frac{2(n-1)}{n}p(y)\,\mathrm{d}y\Big(\int^\infty_0y^\frac{1}{n}p(y)\,\mathrm{d}y\Big)^2>\int^\infty_0y^\frac{2}{n}p(y)\,\mathrm{d}y\Big(\int^\infty_0y^\frac{(n-1)}{n}p(y)\,\mathrm{d}y\Big)^2 \nonumber \end{equation}$

$p(y)>0$ and $\int^\infty_0p(y)\,\mathrm{d}y=1$ for any y

n is an integer and $n\ge2$

 

[sennyk]

When n = 2, both of those inequalities are equal, thus the inequality isn't true for n = 2.