I am working on the following problem.
Given a pdf $f$ that is non-decreasing in the interval $ a \leq x \leq b $, show that for any $s>0$ $\int^b_{a}{x^{2s}f(x)}dx \geq \frac{b^{2s+1}-a^{2s+1}}{(2s+1)(b-a)} \int^b_{a}{f(x)dx}$.
I don't understand. how to start...can any one provide me solution of that problem. Thanks in advance
Hint: For every nondecreasing real-valued functions $u$ and $v$ and every probability measure $\mu$ on the real line $\mathbb R$, $$\int_\mathbb R uv\mathrm d\mu\geqslant\int_\mathbb R u\mathrm d\mu\cdot\int_\mathbb R v\mathrm d\mu.$$