Problem 3

Determine the least real number

M

such that the inequality

\left| ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})\right|\leq M(a^{2}+b^{2}+c^{2})^{2}

holds for all real numbers

a

b

and

c

Step 2 of 8: Order the variables (the bound is symmetric)

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In plain words

Swapping any two of $a$ , $b$ , $c$ only flips the sign of $(b-c)(a-b)(a-c)(a+b+c)$ , and both the absolute value and $(a^2+b^2+c^2)^2$ ignore that sign, so we may fix a convenient order.

a\le b\le c

Detailed analysis

The whole inequality is unchanged by any permutation of $a$ , $b$ , $c$ : the right side $(a^2+b^2+c^2)^2$ is manifestly symmetric, and the left side changes only up to sign under a permutation of the factored form $(b-c)(a-b)(a-c)(a+b+c)$ , which the absolute value erases. Hence we may assume $a\le b\le c$ .