Let’s suppose , , , are non-negative. (Obviously will have to be positive if we want to maximize it.)
Applying AM–GM to the first equation gives
and e is maxed when .
This is consistent with the application of AM–GM to the second equation, when we get
(Clearly, for positive e, e is maxed if and only is maxed.)
Solving those two equations in and gives as the maximum value for .
If some or all of , , , are negative, some other method may have to be tried.