Dear All

I have problem with the following inequality

Let $\displaystyle f = f(x)$ be a nonnegative even function that is strictly increasing for positive $\displaystyle x$. Then for a random variable $\displaystyle \xi$ with $\displaystyle |\xi(\omega)|\leq C$.

$\displaystyle \frac{{Ef(\xi ) - f(\varepsilon )}}{{f(C)}} \le P\left\{ {\left| {\xi - E\xi } \right| \ge \varepsilon } \right\} \le \frac{{Ef(\xi - E\xi )}}{{f(\varepsilon )}}.$

By Markov inequality

$\displaystyle P\{\xi \ge \epsilon\} \le \frac{E\xi}{\epsilon}$

I consider $\displaystyle \eta = |\xi - E\xi|$. Now if we have a function $\displaystyle f: X \rightarrow R^{+}$ then

$\displaystyle P\{f(\eta) \ge f(\epsilon)\} \le \frac{Ef(\eta)}{f(\epsilon)}$

(Is it correct and why)

Now if $\displaystyle f$ is strictly increasing function then

$\displaystyle P\{\eta \ge \epsilon\}=P\{ f(\eta) \ge f(\epsilon)\} \le \frac{Ef(\eta)}{f(\epsilon)}$

So I obtained:

$\displaystyle P\{|\xi - E\xi| \ge \epsilon\} \le \frac{Ef(|\xi - E\xi|)}{f(\epsilon)}$

How can be proved this part.

$\displaystyle \frac{{Ef(\xi ) - f(\varepsilon )}}{{f(C)}} \le P\left\{ {\left| {\xi - E\xi } \right| \ge \varepsilon } \right\}$

Thanx