prove inequality

JennyPA

I found this problem the other day, seems interesting but I am still not sure about the solution
Anybody can help

x, y, z are numbers with

x+y+z=1 and 0<x,y,z<1

prove that

sqrt(xy/(z+xy))+sqrt(yz/(x+yz))+sqrt(xz/(y+xz))<=3/2 ("<=" means less or equal)

mathman

I haven't worked it through. However, it looks like the max is assumed when x=y=z=1/3. In that case each term in the sum = 1/2.

JennyPA

You are right I did that part already and think it's not enough

micromass

What do you mean, it is not enough?? Why do you think that??

JennyPA

It should be proved for all other combinations of x,y and z.
Proving only for x,y and z equal 1/3 is incomplete

micromass

But the expression reaches a maximum when x=y=z=1/3, so doesn't that tell you anything?

checkitagain

Devil's Advocate:

** You claim that the maximum is reached when x = y = z = 1/3, but
it has not been shown by anyone in this thread that it is (or referenced to
another place as already known to be).

So . . . . . I am not convinced that the expression could not be greater than 3/2.


mathman stated that "it looks like the max..." <---- Not concrete

When JennyPA stated that she "did that part already" in response
to mathman, she may have just referred to plugging in 1/3 for each
variable to see that particular sum is 3/2 , but had no knowledge of
whether that selection makes a maximum or not.

mathman

I suggest that someone use some elementary calculus and see of the max is where it appears to be. To avoid partial derivatives, hold x constant and see if the max is at y=z. If that works, the rest should be easy enough.

fbs7

There was a previous discussion on this forum about this hypothesis:

If f(x,y,z) is continuous and symmetric, that is, f(x,y,z) = f(y,z,x) = f(z,x,y), then if that function has a global maximum at f(x*,y*,z*), then x*=y*=z*

That discussion didn't include a proof of that, though.

pwsnafu

IIRC it was a function had to have a unique global maximum.

fbs7

Good point: if it's unique, then obviously x*=y*=z*

coolul007

Just a little algebra and the each term can eliminate a variable: xy/(z + xy) = xy/[(1-x)(1-y)]
Likewise for the other 2 terms.

JennyPA

I am still working on this problem and has no clue how to proceed from here.

Office_Shredder

The best suggestion kind of got lost in the noise here. Take derivatives and use Lagrange multipliers/whatever calculus method you feel most comfortable with to find the maximum of the function

JennyPA

The problem is I am not suppose to use calculus for this problem. This is the algebra course

coolul007

xy/[(1-x)(1-y)] when you eliminate z, then this expression is always 1 when x+y = 1

JennyPA

how you eliminate z????
you can replace z with 1-x-y but can't see what you did