I have some diffculties with the following question; Prove that $ES_{\alpha}(L) = \frac{1}{1- \alpha}inf_{c \in {R}}\{E[(L-c)^{+}] + (1-\alpha)c\}$
Hint use
$ES_{\alpha}(L) = \frac{1}{1-\alpha}E\big[I_{\{F^{-1}(\alpha) \leq L\}} L\big]$
and assume that
$f(c) = E[(L-c)^{+}] + (1 - \alpha)c$
is a differentiable function