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Adele000

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Let x,y,z≥0 x,y,z≥0. Show that x4(y+z)+y4(z+x)+z4(x+y)≤1/12 *(x+y+z)5
 
Adele000 said:
Let x,y,z≥0 x,y,z≥0. Show that x4(y+z)+y4(z+x)+z4(x+y)≤1/12 *(x+y+z)5
Click to expand...
Does your expression look like:

For x,y,z ≥ 0 Show that

x4(y+z) + y4(z+x) + z4(x+y) ≤ 1/12 * (x+y+z)5

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:


Please share your work/thoughts about this problem.
 
I think that I proved the inequality If x=0 by assuming WLOG that y+z=1 and creating a perfect square.But I font know how tot solve it If x>0
 
Adele000 said:
I think that I proved the inequality If x=0 by assuming WLOG that y+z=1 and creating a perfect square.But I font know how tot solve it If x>0
Click to expand...
You say:

I proved the inequality If x=0 by assuming WLOG that y+z=1 and creating a perfect square

Please share your work regarding that proof.
 
When x=0 and find the equality case with y+z=1.
y=1/2−1/2*√3 or 1/2+1/2*√3
I think that now I know how to prove it if x=0 by creating a perfect square.Can you help me please if x>0?
 
Adele000 said:
When x=0 and find the equality case with y+z=1.
y=1/2−1/2*√3 or 1/2+1/2*√3
I think that now I know how to prove it if x=0 by creating a perfect square.Can you help me please if x>0?
Click to expand...
You said

with y+z=1.
y=1/2−1/2*√3 or 1/2+1/2*√3

Please show work.
 
You are supposed to prove something for x,y,z ≥ 0. However you claim that you proved the fact using x=0 and y+z=1. If now you show that it is true for x>0 and y+z = 1 then the proof will still be incomplete. It must be true for ANY x, y, z > 0.
 
Did you try multiplying everything out and cancelling like terms?
 
And what in the world is ”wlog” mean? I don’t think it is an English word. Would you please translate it from Turkish or whatever your native language is?
 
By the way, you can make your life easier by assuming

[MATH]0 \le x \le \le y \le z[/MATH] because everything is symmetric.

I suspect that the proof is relatively trivial unless

[MATH]0 < x < y < z.[/MATH]
So I might work on proving that. Then the more restrictive cases should be easy to add in.
 
Jomo said:
In case you are serious, wlog means without loss of generality
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I never studied the language of the Acronyms or the IMs. I hear both peoples have a fascinating culture.

But it is a little mysterious how specifying x = 0 and y + z = 1 represents no loss of generality when the problem merely specifies that the variable are non-negative.
 
Iam kind of lost this is the first mathematicians I have been able to get in contact with my question or reply has nothing to do with this question it is simply
how many cubic degrees in a 4-hypersphere
 
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