Question 177
Suppose $a$, $b$ and $c$ are positive real numbers. Prove that
$$
\frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \le \frac{\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}}{3}.
$$
For each of the inequalities, find conditions on $a$, $b$ and $c$ such that equality holds.
$$
\frac{a+b+c}{3} \le \sqrt{\frac{a^2+b^2+c^2}{3}} \le \frac{\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b}}{3}.
$$
For each of the inequalities, find conditions on $a$, $b$ and $c$ such that equality holds.