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

Question

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

Uniteasy Admin · Accepted Answer

Using Cauchy–Schwarz and quadratic inequalities
$$a^4+b^4+c^4=2(a^4+b^4+c^4)/2$$
$$=[(a^4+b^4)+(b^4+c^4)+(a^4+c^4)]/2$$
≥$$(2a^2b^2+2b^2c^2+2c^2a^2)/2$$
$$≥a^2b^2+b^2c^2+a^2c^2$$
$$=[a^2(b^2+c^2)+c^2(a^2+b^2)+b^2(a^2+c^2)]/2$$
$$≥a^2bc+c^2ab+b^2ac≥abc(a+c+b)$$

Question

Answer