I would need, if possible, some help with the calculation of the Orlicz norm of a random variable $X$.
I have this function:
$\phi(x)=x\ln(1+x)$,
and I need to compute
$\inf c>0$ s.t. $E \left[\phi\left(\frac{|X|}{c}\right) \right]\leq 1$.
I cannot find a solution for this (sorry...), so any help will be gratefully appreciated.