Prove that for all positive reals $a, b, c$ the following double inequality holds: $$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$
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Tags: 3rd edition, inequalities, algebra
Prove that for all positive reals $a, b, c$ the following double inequality holds: $$\frac{a+b+c}{3}\ge \sqrt[3]{\frac{(a+b)(b+c)(c+a)}{8}}\ge \frac{\sqrt{ab}+\sqrt{bc}\sqrt{ca}}{3}$$