Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

Topic: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Replies: 20   Last Post: Nov 1, 2013 4:48 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
Don Coppersmith

Posts: 43
Registered: 2/2/06
Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Posted: Oct 25, 2013 1:58 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

A = sqrt(3).
Assume x,y,z nonnegative.

Let T = x^2+y^2+z^2,
S = x+y+z,
P = xyz.

Express T^2-3SP as the sum of squares:
T^2-3SP = (x^2-yz)^2 + (y^2-zx)^2 + (z^2-xy)^2
+ (1/2)[(xz-yz)^2 + (xy-yz)^2 + (xy-xz)^2]

Then T^2-3SP >= 0,
T^2 >= 3SP >= 3P^2
T >= sqrt(3)P
An extremal example is x=y=z=sqrt(3).

Don Coppersmith

> Here's a nice challenge problem which I adapted from
> a past
> competition problem ...
>
> Problem:
>
> Find, with proof, the largest real number A such that
>
> x + y + z >= xyz
>
> implies
>
> x^2 + y^2 + z^2 >= Axyz
>
> My solution is elementary (avoids Calculus), but if a
> method
> based on Calculus yields an easy resolution, that
> would be worth
> seeing as well.
>
> quasi



Date Subject Author
10/25/13
Read x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
David C. Ullrich
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/26/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Karl-Olav Nyberg
10/26/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/26/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Peter Percival
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
David C. Ullrich
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Don Coppersmith
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Don Coppersmith
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
gnasher729
10/25/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Peter Percival
10/27/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
gnasher729
10/27/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
dan.ms.chaos@gmail.com
10/27/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Peter Percival
10/27/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
fom
10/30/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/30/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Peter Percival
10/30/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/31/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
10/31/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
quasi
11/1/13
Read Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Phil Carmody

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.