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 Posted: Mon, 14 Jan 2008 10:23:45 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 18 Mar 2006 08:42:24 UTC
Posts: 834
Challenge other members if you have interesting problems in Inequalities or Integration!

You do not need to know the solution yourself!

Please do NOT post easy, boring, or IRRELEVANT problems in this thread!

I will post my problems later, but first I'd like to see some activities from other members! Let's see what you've got!

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 Post subject: Inequalities 1, 2Posted: Mon, 14 Jan 2008 14:27:30 UTC
 S.O.S. Oldtimer

Joined: Tue, 20 May 2003 17:25:56 UTC
Posts: 235
Location: Israel
Hello. Here is an inequality(1) (not proved by me yet, but I think it's proved by someone else, i.e. it's not a conjecture).

Prove that for all the following holds:

Another inequality (2). (This one is known)
Let a, b and c be positive reals. Prove that

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 Post subject: Re: Inequalities 1, 2Posted: Tue, 15 Jan 2008 17:08:16 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 18 Mar 2006 08:42:24 UTC
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kar wrote:
Prove that for all the following holds:

this is a very strange result! i've found a few lower bounds for but all of them are annoyingly smaller
than i used a strong version of Stirling's approximation but that made things even more

complicated. i have no idea how to prove this! anyone?

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 Post subject: Posted: Sun, 27 Jan 2008 18:32:13 UTC
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Problem 1 Easy! I've mentioned it earlier in Algebra forum:

Show that for all Find all for which equality holds.

Problem 2 Hard I guess, and I don't know the answer:
For a given natural number n and integers (or more generally reals) , find .

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 Post subject: Posted: Tue, 29 Jan 2008 09:03:38 UTC
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Joined: Sun, 4 May 2003 16:04:19 UTC
Posts: 2906
This is supposedly made by me (I don't remember).
suppose a,b,c are positive reals such that abc=1
show:

_________________
Has anyone noticed that the below is WRONG? Otherwise this statement would be true:

where

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 Post subject: Posted: Mon, 4 Feb 2008 12:36:39 UTC
 S.O.S. Oldtimer

Joined: Wed, 30 Aug 2006 00:39:19 UTC
Posts: 243
Try this problem:
Prove that

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The shortest path between two truths in the real domain passes through the complex domain. - Jacques Hadamard

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 Post subject: Posted: Tue, 5 Feb 2008 03:48:54 UTC
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bidentate wrote:
Try this problem: Prove that

it's quite easy: let then
therefore:
, and hence

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 Post subject: Posted: Wed, 6 Feb 2008 09:20:57 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 18 Mar 2006 08:42:24 UTC
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Prove, by induction, that for all integers

Note: For a non-inductive proof of this inequality see this thread.

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 Post subject: Posted: Mon, 11 Feb 2008 20:28:02 UTC
 S.O.S. Oldtimer

Joined: Thu, 13 Jul 2006 19:47:54 UTC
Posts: 308
How about a problem that is both an inequality and an integral?

Let f(x) be a function such that .

Prove that and only equals 0 when almost everywhere.

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 Post subject: Posted: Mon, 11 Feb 2008 22:26:35 UTC
 Moderator

Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15563
Location: Austin, TX
Zathras wrote:
How about a problem that is both an inequality and an integral?

Let f(x) be a function such that .

Prove that and only equals 0 when almost everywhere.

By almost everywhere, do you mean off a set of measure 0?

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 Post subject: Posted: Mon, 11 Feb 2008 23:24:35 UTC
 S.O.S. Oldtimer

Joined: Thu, 13 Jul 2006 19:47:54 UTC
Posts: 308
Zathras wrote:
How about a problem that is both an inequality and an integral?

Let f(x) be a function such that .

Prove that and only equals 0 when almost everywhere.

By almost everywhere, do you mean off a set of measure 0?

Yes.

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 Post subject: Posted: Tue, 12 Feb 2008 01:29:40 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Sat, 18 Mar 2006 08:42:24 UTC
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Zathras wrote:
How about a problem that is both an inequality and an integral?

Let f(x) be a function such that .

Prove that and only equals 0 when almost everywhere.

let then , and thus ,

because , for all the second part of your problem also follows quickly: since ,

for all x, we have , iff almost everywhere iff almost everywhere. Q.E.D.

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 Post subject: Posted: Fri, 15 Feb 2008 09:10:24 UTC
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For any integrable function :

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 Post subject: Posted: Sun, 17 Feb 2008 02:38:31 UTC
 Senior Member

Joined: Mon, 12 Nov 2007 00:25:37 UTC
Posts: 106

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 Post subject: Posted: Sun, 17 Feb 2008 21:43:51 UTC
 Member

Joined: Sat, 16 Feb 2008 21:22:30 UTC
Posts: 22

Are the denominators of the 2nd and 3rd expression purposely equal or is it a mistake?

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