How can I show monotonicity of this function defined on the space of random variables $RV(\Omega)$? I have the function $f_{\lambda}:RV(\Omega)\rightarrow \mathbb{R}$ defined on the space $RV(\Omega)$ supported over some scenario set $\Omega$:
$f_{\lambda}:=\frac{1}{\lambda}\log(\mathbb{E}[e^{-\lambda X}])]$ ,
where $\lambda>0$. Now, I want to show that this function is monotone. In order to do this I think I have to show that for any $X,Y\in RV({\Omega})$ with $X(\omega)\leq Y(\omega)$ $\forall\omega\in\Omega$ I have that:
$f_{\lambda}(X)\geq f_{\lambda}(Y)$
However, I am not sure how to show that this inequality holds. Any ideas?

Note that
\begin{align}
f_\lambda(X)-f_\lambda(Y) =&\frac{1}{\lambda}\log \left( \frac{\mathbb{E}(e^{-\lambda X})}{\mathbb{E}(e^{-\lambda Y})}\right) \\
\geq & \frac{1}{\lambda} \log 1 \geq 0 \, ,
\end{align}
where in the penultimate inequality I have used the fact that $X \leq Y$ and $\lambda>0$ implies that $e^{-\lambda X}\geq e^{-\lambda Y}$ for all $\omega \in \Omega$.
