For which a,b,ca, b, c does the inequality a2b+ab2+b2c+bc2+c2a+ca26abca^2b+ab^2+b^2c+bc^2+c^2a+ca^2\ge 6abc hold?

I know that it holds for non-negatives a,b,ca, b, c but consider the following:

The inequality a3+b3+c33abca^3+b^3+c^3\ge 3abc holds for non-negatives a,b,ca, b, c but it holds for real numbers as well iff (a+b+c)0(a+b+c)\ge 0 since: a2+b2+c2abbcca0a,b,cRa+b+c0(a+b+c)(a2+b2+c2acbcca)0Euler’s Formulaa3+b3+c33abc0a3+b3+c33abca^2+b^2+c^2-ab-bc-ca\ge 0 \quad \forall a, b, c \in \mathbb{R} \\ \stackrel{a+b+c\ge 0}{\Longleftrightarrow} (a+b+c)(a^2+b^2+c^2-ac-bc-ca)\ge 0 \\ \stackrel{\text{Euler's Formula}}{\Longleftrightarrow} a^3+b^3+c^3-3abc\ge 0 \\ \Leftrightarrow a^3+b^3+c^3\ge 3abc (Generally Note that we can prove the same way that a+b+c0a3+b3+c33abca+b+c\le 0 \Leftrightarrow a^3+b^3+c^3\le 3abc)

Considering that the inequality 2(a3+b3+c3)a2b+ab2+b2c+bc2+c2a+ca22(a^3+b^3+c^3)\ge a^2b+ab^2+b^2c+bc^2+c^2a+ca^2 holds for all real numbers by rearrangement Inequality, I suppose that something similar for the inequality a2b+ab2+b2c+bc2+c2a+ca26abc()a^2b+ab^2+b^2c+bc^2+c^2a+ca^2\ge 6abc \quad (*) should exist. But unfortunately (a+b+c)0(a+b+c)\ge 0 is not the condition we are looking for...

I don't really know if that helps but ()(a+b)(b+c)(c+a)8abc(a+b+c)(ab+bc+ca)9abc(*) \Leftrightarrow (a+b)(b+c)(c+a)\ge 8abc \Leftrightarrow (a+b+c)(ab+bc+ca)\ge 9abc

Note by Chris Galanis
3 years, 2 months ago

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