Let $X$, $Y$ some non-negative random variables and let $X\leq_{Lt}Y$ that means $E[\exp(-tX)]\geq E[\exp(-tY)]$ for all $t>0$. And let $\pi_{\operatorname{Exp}}(X)=\frac{1}{\alpha}\ln(E[\exp(\alpha X)])$ for some $\alpha>0$ the Exponential premium. Than I have read in some article, that $X\leq_{Lt}Y$ implies $\pi_{\operatorname{Exp}}(X)\geq\pi_{\operatorname{Exp}}(Y)$, so it must hold that $E[\exp(\alpha X)]\geq E[\exp(\alpha Y)]$.
How can I show this?