The sum of three square roots bounded below by $\sqrt{82}$ 
Let $a,b,c >0$ and $a+b+c \le  1$. Prove: $$\sqrt{a^2+1/a^2}+\sqrt{b^2+1/b^2}+\sqrt{c^2+1/c^2} \ge \sqrt{82}$$

Progress: I tried to have 3 vectors $(a,1/a)$, $(b,1/b)$ and $(c,1/c)$, played around with the vector inequalities but no success.

It can be seen that $\sqrt{x^2+\frac{1}{x^2}}$ is convex. Therefore we can use Jensen inequality: 
$$
\frac{1}{3}\sqrt{a^2+1/a^2}+\frac{1}{3}\sqrt{b^2+1/b^2}+\frac{1}{3}\sqrt{c^2+1/c^2} \ge \sqrt{(\frac{a+b+c}{3})^2+(\frac{3}{a+b+c})^2}.
$$
On the other hand because $a+b+c<1$, we can see that $f(x)=\sqrt{\frac{x^2}{9}+\frac{9}{x^2}}$ for $x<1$ has its minimum at $x=1$ and hence:
$$
\frac{1}{3}\sqrt{a^2+1/a^2}+\frac{1}{3}\sqrt{b^2+1/b^2}+\frac{1}{3}\sqrt{c^2+1/c^2} \ge \sqrt{\frac{82}{9}}.
$$
which proves the inequality.
