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}<br /> \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&gt;\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<br /> \nonumber<br /> \end{equation}

p(y)&gt;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.