Question

Show that a^4+b^4+c^4≥ 2b^2+b^2c^2+c^2a^2≥(a+b+c)abc

Collected in the board: Inequality

Steven Zheng posted 12 hours ago

Answer

Using Cauchy–Schwarz and quadratic inequalities

a^4+b^4+c^4=2(a^4+b^4+c^4)/2

=[(a^4+b^4)+(b^4+c^4)+(a^4+c^4)]/2

(2a^2b^2+2b^2c^2+2c^2a^2)/2

≥a^2b^2+b^2c^2+a^2c^2

=[a^2(b^2+c^2)+c^2(a^2+b^2)+b^2(a^2+c^2)]/2

≥a^2bc+c^2ab+b^2ac≥abc(a+c+b)

Steven Zheng posted 12 hours ago

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