Jensen's inequality and Cauchy

Silverfalcon

I heard of those two inequalities.

But what are they? Confused

**Posted:** Mon Feb 21, 2005 12:07 pm

blahblahblah

I posted a thread on Cauchy and its proof in the Pre-Olympiad forum not too long ago.

As for Jensen's, I can tell you what it is, but unless you know calculus and understand the geometrical interpretation of certain things in calculus, there's not much point to me telling you.

If you want to get started in inequalities, start with rearrangement and AM-GM, imho.

**Posted:** Mon Feb 21, 2005 12:12 pm

Rep123max

Rearrangement? I've never heard of that. Is that like a name for a more common thing that I probably know?

**Posted:** Mon Feb 21, 2005 2:29 pm

paladin8

Concisely, given two monotonically increasing sequences $a$ and $b$ , $\displaystyle \sum{a_ib_i} \ge \sum{a_ib_{\delta(i)}}$ where $\delta$ is a permutation of the $i$ 's.

**Posted:** Mon Feb 21, 2005 3:28 pm

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joml88

Strangely enough, rearrangement isn't covered in either AoPS V2 or Zeitz... I found out about it here (go to Inequalities (II).)

**Posted:** Mon Feb 21, 2005 5:09 pm

tetrahedr0n

Its covered well in Kiran Kedlaya's a>b.

**Posted:** Mon Feb 21, 2005 6:09 pm

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Magnara

Rearrangement is my new favorite inequality, by far. I just learned about it last week. Engel loves it.

**Posted:** Mon Feb 21, 2005 6:54 pm

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tetrahedr0n

Yes, Engel even does Rearrangement for more than two sequencs, which I haven't seen anywhere else, but does help on some problems.

**Posted:** Tue Feb 22, 2005 1:49 pm

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