Maximum of two symbolic expressions in Mathematica - wolfram-mathematica

Question

If I have two symbolic expressions and the algebra can tell which one is the largest, how can I get Mathematica to find that out? Example:
A = s1/z1 + s2/z2 + Sqrt[(s1/z1 + s2/z2)^2 + 4 (z1^2 z2^2 - s1 s2)/(z1 z2)];
B = s1/z1 + s2/z2 - Sqrt[(s1/z1 + s2/z2)^2 + 4 (z1^2 z2^2 - s1 s2)/(z1 z2)];
Assuming[s1 > 0 && s2 > 0 && z1 > 0 && z2 > 0, Max[A, B]]
This gives
Max[s1/z1 + s2/z2 - Sqrt[(s1/z1 + s2/z2)^2 + (4 (-s1 s2 + z1^2 z2^2))/(z1 z2)],
s1/z1 + s2/z2 + Sqrt[(s1/z1 + s2/z2)^2 + (4 (-s1 s2 + z1^2 z2^2))/(z1 z2)]]
which not much of use.
However, one can see that it's always A < B, because A is a difference between two positive numbers, while B is the sum. I'd like Mathematica to tell me that the Max is always B. Can it be done?

Answer 1

had a chance to check ..
Assuming[ {z1 > 0, z2 > 0, s1 > 0, s2 > 0} , Simplify[Reduce[A > B]]]
-> True
note by removing the Simplify and looking at the conditional expression produced by Reduce you can see how to relax the assumptions a bit:
Assuming[ {z1 > 0, z2 > 0, Element[{s1, s2}, Reals]} , Simplify[Reduce[A > B]]]
-> True
Note also Reduce[A < B] returns False with no assumptions. That expression can not be true even with complex expressions.
I suppose if you want to implement something like Max you could do this:
Assuming[ {z1 > 0, z2 > 0, s1 > 0, s2 > 0} ,
Which[ Simplify[Reduce[A > B]] , A ,
Simplify[Reduce[B > A ]] , B ,
True , Indeterminate ]]

Answer 2

The answer by agentp can be also made iterative, so one can generalize this to more than two expressions:
v = {A, B};
max = B;
Assuming[{z1 > 0, z2 > 0, s1 > 0, s2 > 0},
For[i = 1, i <= 2, i++,
If[Simplify[Reduce[v[[i]] > max]], max = v[[i]]]
]
]
Print[max];
-> s1/z1+s2/z2+Sqrt[(s1/z1+s2/z2)^2+(4 (-s1 s2+z1^2 z2^2))/(z1 z2)]