I tried to solve this inequality because I find exotic. Actually, I didn't look at the right solution. Because before I look at the right solution, I want to know if my solution is right or not.
Prove that $0 ≤ yz + zx +xy -2xyz≤\frac{7}{27}$, where $x,y$ and $z$ are non-negative real numbers for which $x + y + z = 1.$
Attempts:
If $x=0$. The left side of the inequality is correct. So, I can accept $x,y,z≠0$.
It is enough to prove $\frac 1x+\frac1y+\frac1z≥2$
We have,
$x+y+z≥3\sqrt[3]{xyz}\Longrightarrow xyz≤\frac1{27}$
$\frac 1x+\frac1y+\frac1z≥\frac3{\sqrt[3]{xyz}}≥9≥2$
The left side proved.
It is obvious at least one of the numbers is less than $\frac 12.$
So, we can choose $y$, such that $y≤\frac 12$.
Therefore, we have
$yz + zx +xy −2xyz ≤ \frac7{27}$
$x(y+z)+yz(1-2x)-\frac7{27}≤0$
$x(1-x)+y(1-x-y)(1-2x)-\frac7{27}≤0$
$x-x^2+y-xy-y^2-2xy+2x^2y+2xy^2-\frac7{27}≤0$
$x^2-x-y+xy+y^2+2xy-2x^2y-2xy^2+\frac{7}{27}≥0$
$x^2(1-2y)+x(3y-2y^2-1)+(y^2-y+\frac7{27})≥0$
$2(\frac12-y)\left(x+\frac{y-1}{2}\right)^2+\frac{1}{108} (3y-1)^2(6y+1)≥0$
Of course, I'm not sure the solution is correct. Can you verify the solution?
Thank you.
Best Answer
I checked your solution. Your solution is right.
I like the following way.
The homogenization helps.
By AM-GM $$xy+xz+yz-2xyz=(x+y+z)(xy+xz+yz)-2xyz\geq9xyz-2xyz=7xyz\geq0.$$ Also, $$xy+xz+yz-2xyz\leq\frac{7}{27}$$ it's $$(xy+xz+yz)(x+y+z)-2xyz\leq\frac{7}{27}(x+y+z)^3$$ or $$\sum_{cyc}(7x^3-6x^2y-6x^2z+5xyz)\geq0,$$ which is true by Schur and AM-GM.