How does Cauchy's inequality work?

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# How does Cauchy's inequality work?

Submitted by caroline on Thu, 04/22/2004 - 02:48

Forums:

cauchy's inequality is supposed to be:

(xy^3+yz^3+zx^3)(z+x+y) *is greater than or equal to* xyz(y+z+x)^2.

i don't get what it's supposed to be used for, and how it works. it just confuses me. please help!!!

thanks!!!

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## Re: How does Cauchy's inequality work?

Hi, it is the squared special instance n=3, x,y,z>=0,

(x1,x2,x3)=(sqrt(xy^3),sqrt(yz^3),sqrt(zx^3)),

(y1,y2,y3)=(sqrt( z ),sqrt( x ),sqrt( y )),

x*y =(sqrt(xyz)y,sqrt(xyz)z,sqrt(xyz)x),

x.y = sqrt(xyz)(y + z + x).

Given x=0, z=1, the inequality reads

y(1+y)>=0 and fails for 0>y>-1, so nonnegativity is necessary.

Greetings,

Andy.

## Re: How does Cauchy's inequality work?

Hi,

For Cauchy inequality see

http://planetmath.org/encyclopedia/CauchyInequality.html

## Re: How does Cauchy's inequality work?

(x+y+z)\neq 0 it is innecesary and (x+y+z)=0 is trivial.