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 5 of 8: Combine the two bounds into a single estimate

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|(b-c)(a-b)(a-c)(a+b+c)|\le\frac14|c-a|^3|a+b+c|\le\frac14\left(\frac{2u}{3}\right)^{3/2}s,\quad u=(b-a)^2+(c-b)^2+(c-a)^2,\ s=|a+b+c|

Detailed analysis

Multiplying the Step 3 bound by $|a-c||a+b+c|$ gives $|(b-c)(a-b)(a-c)(a+b+c)|\le\frac14|c-a|^3|a+b+c|$ . Writing $u=(b-a)^2+(c-b)^2+(c-a)^2$ and $s=|a+b+c|$ , Step 4 gives $|c-a|^2\le\frac{2u}{3}$ , hence $|c-a|^3\le\left(\frac{2u}{3}\right)^{3/2}$ and the bound becomes $|(b-c)(a-b)(a-c)(a+b+c)|\le\frac14\left(\frac{2u}{3}\right)^{3/2}s$ .