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

andeis maxed when .

This is consistent with the application of AM–GM to the second equation, when we get

when .

(Clearly, for positivee,eis 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.