(July 31, 2012 at 6:18 pm)Categories+Sheaves Wrote: 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
Which is true by the concavity of f(x) = xg for g in [0,1]
You've improperly switched some negatives to positives there, I think.
Quote:#2 is simple if you interpret each side as an integral of xg's derivative (which is monotonically decreasing, etc.).
Anyway, that's the easy route. Since you've specified that this is a simple real analysis question, were you looking for a purely algebraic argument? (because it feels like there should be one...)
Any proof will do, it's a useful lemma for my girlfriend's research.
Quote:afterthought: you meant "let a,b be integers", right?
Yeah, I messed with the formatting and forgot to change my variable names.
“The truth of our faith becomes a matter of ridicule among the infidels if any Catholic, not gifted with the necessary scientific learning, presents as dogma what scientific scrutiny shows to be false.”
