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 Post subject: transcendental inequalitiesPosted: Sun, 25 Oct 2009 21:36:15 UTC
 Senior Member

Joined: Mon, 29 Jun 2009 00:59:53 UTC
Posts: 126
Location: some space-time singularity
Prove following 3 inequalities:

1)

2)

3)

I believe all 3 are genuine mine, but if someone thinks he has already seen them elsewhere, please point to the source. I would be interested in having a look into it. Also, it would be nice to see if there are different solutions to these problems.

Good luck and have fun

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 Post subject: Re: transcendental inequalitiesPosted: Sun, 25 Oct 2009 23:22:23 UTC
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Joined: Wed, 30 Mar 2005 04:25:14 UTC
Posts: 15563
Location: Austin, TX
ex falso quodlibet wrote:
Prove following 3 inequalities:

1)

2)

3)

I believe all 3 are genuine mine, but if someone thinks he has already seen them elsewhere, please point to the source. I would be interested in having a look into it. Also, it would be nice to see if there are different solutions to these problems.

Good luck and have fun

2)

Spoiler:
Let

Then this reduces to:

Applying the half-angle identity

now bring the under the square root in the middle term.

This is obviously true if we can get that for every

since all terms are positive, squaring preserves ordering, so examine the resultant expression:

this is true iff

Now the result is obvious, because the is an even function, so has all even terms in its taylor series, and the second order approximation is , in particular it is evident that the inequality is true, hence--as each step was reversible, the argument shows the inequality to be true.

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