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Use the Rearrangement Inequality to show that...

(1) For nonnegative a,b,c, [unparseable or potentially dangerous latex formula]

(2) For positive a,b,c, [unparseable or potentially dangerous latex formula]

(3) [unparseable or potentially dangerous latex formula]

(4) For all a,b,c, [unparseable or potentially dangerous latex formula]


From Engel Ch. 7.

I can do number 1, but I don't know how to use LaTeX and it'd be too complicated without it.

Sorry, misread #1, but i can do #2:

Assume that [unparseable or potentially dangerous latex formula] so that [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are similarly arranged.

Thus [unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]

Adding these together gives us:

[unparseable or potentially dangerous latex formula]

The second part of this inequality gives us 3 (proof to long to type out), and then upon dividing both sides by 2 you get

[unparseable or potentially dangerous latex formula]

it appears #1 was supposed to say nonnegative numbers. Say, [unparseable or potentially dangerous latex formula], then [unparseable or potentially dangerous latex formula], which is clearly untrue. Sorry, 'bout that. You should be able to do it now.

1) very trivial but what the heck

WLOG assume [unparseable or potentially dangerous latex formula] then [unparseable or potentially dangerous latex formula]

So then by rearrangement [unparseable or potentially dangerous latex formula] as desired

for 2) i did a combination of Cauchy and Rearrangment, if you want to see my solution tell me

I'm assuming that if I got #3 without using Rearrangement Inequality, then it's cheating

#2 is Nesbitt's Inequality

I believe the standard approach is to add three to both sides:
[unparseable or potentially dangerous latex formula]
If the first factor, [unparseable or potentially dangerous latex formula] is rewritten as [unparseable or potentially dangerous latex formula], then either Cauchy-Schwartz or Rearrangement (Chebyshev, as [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are oppositely sorted) will do the trick:
[unparseable or potentially dangerous latex formula]

#4. Although there is mention of some scalar Rearrangement form, if I remember correctly, Engel never proves it, so we'll resort to other methods first, such as the Trivial Inequality:
[unparseable or potentially dangerous latex formula]
[unparseable or potentially dangerous latex formula]
We can use the Rearrangement Inequality (or many others) to show that
[unparseable or potentially dangerous latex formula]
By transitivity, [unparseable or potentially dangerous latex formula] and the desired result follows.