a^2b+ab^2-2abc++b^2c+bc^2 -2abc+c^2a+ca^2-2abc =a^2b+bc^2-2abc+ab^2+ac^2 -2abc+b^2c+ca^2-2abc =b(a^2+c^2-2ac)+a(b^2+c^2-2bc) +c(a^2+b^2-2ab) =b(a-c)^2+a(b-c)^2+c(a-b)^2 >0
$a^b+ab^2-2abc+b^c+bc^2-2abc+
a^c
+ac^2-2abc
$$\Leftrightarrow a^2b+b^2
c-2abc+ab^2+ac^2-2abc+b^2c+ca^2-2abc
$$<=
>b(a-c)^2+a(b-c)^2+c(a-b)^2
\geq 0
$