I think the + and × are backwards from the way that would be helpful. But if you think it would work, try writing it out in detail.
1. Take the log of all terms (allowed, because log keeps the sum monotonic):
log(sqrt(a)) + log(sqrt(b)) + log(sqrt(c))
2. pull out the sqrt from the log:
1/2 * log(a) + 1/2 * log(b) +1/2 * log(c)
3. factor out the 1/2:
1/2 * (log(a) + log(b) + log(c))
4. sum of logs can be rewritten as a log of product:
1/2 * log (a * b * c)
5. compute log of (a * b * c), and halve it. Ditto with log of (d * e * f). This should give a number which is proportional to the original sum of sqrt.
It seems you're employing a + b < c <=> log(a) + log(b) < log(c), which doesn't hold (consider 10, 10, and 20).
(the real rule is a * b < c <=> log(a) + log(b) < log(c)), because log(a) + log(b) <=> log(a * b)
> 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