Hello,

I have to show the following assertion:

$\displaystyle \sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]=0$

I know that

$\displaystyle E[Z]\leq 0$ and $\displaystyle \sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]\geq 0$

and for $\displaystyle a=1$ we have $\displaystyle \sup\limits_{a \in [0,1]} E[log(1+a(exp(Z)-1))]=E[Z]$

Unfortunately I have no idea how to show it.

Does anyone have a hint?

Thanks in advance!