Some kind of average A(a,b) of two numbers a and b should at least:
i) always be a value between a & b
ii) satisfy A(a,b) =A(b,a)
iii) satisfy A(ka,kb)=kA(a,b).
If A(a,b) behaves like an average, must f^(-1)(A(f(a),f(b)) behave like one too for any invertible function f?