This doesn't work. Even if you assume sqrt(a) + sqrt(b) > sqrt(c) + sqrt(d), that doesn't necessarily mean that log(sqrt(a)) + log(sqrt(b)) > log(sqrt(c)) + log

This doesn't work. Even if you assume sqrt(a) + sqrt(b) > sqrt(c) + sqrt(d), that doesn't necessarily mean that log(sqrt(a)) + log(sqrt(b)) > log(sqrt(c)) + log(sqrt(d)). For example, let a = 1, b = 100, c = 25, d = 25. Then > sqrt(1) + sqrt(100) 11.0 > sqrt(25) + sqrt(25) 10.0 > log(sqrt(1)) + log(sqrt(100)) 2.302585092994046 > log(sqrt(25)) + log(sqrt(25)) 3.2188758248682006