RE: Need a proof (real analysis)
July 31, 2012 at 6:18 pm
(This post was last modified: July 31, 2012 at 6:23 pm by Categories+Sheaves.)
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]
#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...)
afterthought: you meant "let a,b be integers", right?
(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]
#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...)
afterthought: you meant "let a,b be integers", right?
