`(x-y)^2+(y-z)^2+(z-x)^2≥0`
`⇔2x^2+2y^2+2z^2-2xy-2yz-2xz≥0`
`⇔x^2+y^2+z^2≥xy+yz+xz`
`⇔(x+y+z)^2≥3(xy+zx+yz)`
thay `x=a^2;y=b^2;z=c^2`
`⇔(a^2+b^2+c^2)^2≥3(a^2b^2+c^2a^2+b^2+c^2)`
`⇔1≥3(a^2b^2+c^2a^2+b^2+c^2)`
`⇔1/(a^2+b^2+c^2+(ac)^2+(bc)^2+(ab)^2)≥3/4`
`a/(1+b^2)+(b)/(1+c^2)+c/(1+a^2)=(a^3)/(a^2+a^2b^2)+(b^3)/(b^2+b^2c^2)+(c^2)/(c^2+c^2a^2)≥(a√a+b√b+c√c)^2/(a^2b^2+c^2a^2+b^2+c^2)=3/4 (a√a+b√b+c√c)^2`
`''=''`xẩy ra khi :
`a=b=c=1/(√3)`