find the value of M such that ab(a^2-b^2)+ b c(b^2-c^2)+ca(c^2-a^2)<=M(a^2+b^2+c

Question

find the value of M such that
ab(a^2-b^2)+ b c(b^2-c^2)+ca(c^2-a^2)<=M(a^2+b^2+c^2)^2 for all a,b,c ?

Explanation / Answer

So in other words, M can be re-written as M>=(a b (a^2-b^2)+b c (b^2-c^2)+c a (c^2-a^2))/(a^2+b^2+c^2)^2 and simplified, M>=((a-b) (a-c) (b-c) (a+b+c))/(a^2+b^2+c^2)^2 or expanded, M>=(a c^3)/(a^2+b^2+c^2)^2-(b c^3)/(a^2+b^2+c^2)^2-(a b^3)/(a^2+b^2+c^2)^2+(b^3 c)/(a^2+b^2+c^2)^2+(a^3 b)/(a^2+b^2+c^2)^2-(a^3 c)/(a^2+b^2+c^2)^2 but unless there's some constraint, M can't be more specific. a>0, M>=(a^3 b-a^3 c-a b^3+a c^3+b^3 c-b c^3)/(a^2+b^2+c^2)^2 a=(a^3 b-a^3 c-a b^3+a c^3+b^3 c-b c^3)/(a^2+b^2+c^2)^2 a = 0, b>0, M>=(b^3 c-b c^3)/(b^2+c^2)^2 a = 0, b=(b^3 c-b c^3)/(b^2+c^2)^2 a = 0, b = 0, c>0, M>=0 a = 0, b = 0, c=0 Hope this helps.

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