Need a proof (real analysis)

[Categories+Sheaves]

[quote='Categories+Sheaves' pid='317718' dateline='1343773102']
You can do the first conjecture in one line by multiplying both sides by (1/3)g, yielding
(a/3)g + (b/3)g +(1/3)g >= (a/3 + b/3 + 1/3)g

You've improperly switched some negatives to positives there, I think.

Ah. Indeed I did.
So that gives us (a-1)g + (b)g >= (a + b - 1)g as long as a > 1 (negatives makes this harder :/ ).
By conjecture #2...
1-0 >= ag - (a-1)g and so (a-1)g >= ag - 1
So just using concavity isn't enough...

Any proof will do, it's a useful lemma for my girlfriend's research.

Because there's nothing more interesting than another person's question: what's the problem/literature you're looking at?