probability inequality

#1
Let X be a random variable such that |X|<=C (bounded). Let f be an even, non decreasing function over the positive values of x. Prove that:

[E(f(X))-f(a)]/f(C) <= P(|X-E(X)|>=a) <= [E(f(X-E(X))]/f(a).

The 2nd inequality is really easy to prove and it follows from chebyshev inequality...I am having hard time solving the first 1, any help is strongly appreciated. Thanks!
 

fed1

TS Contributor
#2
Let a = 0.
Let f(s) = s^2.
Let c = 10.
let Px(x) = Pr[X = x] = u for X >0, l = 1 - u;

Pr[ |X - u| >= a ] = 1.

E[ x^2/100 ] = integral( x^2 * l) from -10 to 0 + integral( x^2 * u) from 0 to 10.

= (1/3)(1/100)(10^3)[ u - l ];

Looks problematic?

Maybe I made algebra mistake?