Max of $P=(a^{2}-ab+b^{2})(b^{2}-bc+c^{2})(c^{2}-ca+a^{2}).$

 Problem. Let $a,b,c\geqslant 0$ and $a+b+c=3$. Find maximum of: 

$$P=(a^{2}-ab+b^{2})(b^{2}-bc+c^{2})(c^{2}-ca+a^{2})$$

Solution. $\text{We can prove:}$

$$\prod(a^2-ab+b^2)+{\frac {1}{243}} \left( a+b+c \right) ^{2} \big[ 2(a^2+b^2+c^2)-5(ab+bc+ca) \big]^{2} \leqslant {\dfrac {4}{243}} ( a+b+c ) ^{6}$$

$\text{Let} $$f(a,b,c)=\text{RHS}-\text{RHS}.$

$\text{Because:}$

• $$f(0,b,c) =\dfrac{1}{27} bc \left( 2b-c \right) ^{2} \left( b-2c \right) ^{2} \geqslant 0.$$
• $$f(a,1,1) ={\frac {8}{27}}{a}^{5}+{\frac {5}{27}}\,{a}^{4}+1/27+{\frac {2}{27}} \,a \left( 49\,{a}^{2}-13\,a+65 \right)\geqslant 0.$$

So we are done.

Note. Method: