Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

pozitive real numbers

Let

be pozitive real numbers satisfying

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: pozitive real numbers

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: pozitive real numbers

Hi Jane;

Thanks for the link.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

tony123

Member

Registered: 2007-08-03

Posts: 229

Re: pozitive real numbers

By AM–GM, we have

and

Also, note that

is positive (because, by AM–GM,

So, multiplying the above two inequalities gives

Hence

Similarly,

and

(by inequality of arithmetic and harmonic means)[/indent]

Multiplying through by

will give the required inequality.

Last edited by tony123 (2009-07-10 21:13:05)