I'm solving some exercises and got stuck. The setting:
let $\varepsilon > 0 $ given, and D(\varepsilon):=\{(x,y) \in \mathbb{R_+}^2\mid |x-y| \ge \varepsilon\mbox{ and }\min(x,y) \le \frac{1}{\varepsilon}\}. Define $ h(x):=\exp{(-x)} $ with domain $ \mathbb{R_+} $. I've proved, that there's a $ \delta > 0 $ (depending on $\varepsilon $ )
$ h\left(\frac{x+y}2\right) \le \frac{h(x)+h(y)}2 - \delta 1_{D(\varepsilon)}(x,y).$
Now how could I use this to show:
$ E(|h(X)-h(Y)|)\le \varepsilon + 2 h\left(\frac{1}{\varepsilon}\right)+ P((X,Y)\in D(\varepsilon))$
where $X,Y$ non negative random variables.
I'm very thankful for any help.