$\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$ is true?

Question

In one of the solutions of a problem in this site: https://artofproblemsolving.com/wiki/index.php?title=2001_IMO_Problems/Problem_2

It is used the following:

If $a,b,c$ are positive real numbers such that $a+b+c = 1$, then

$$\frac{a}{\sqrt{a^2 + 8bc}} +\frac{b}{\sqrt{b^2 + 8ac}} + \frac{c}{\sqrt{c^2 + 8ab}} \ge \frac{1}{\sqrt{a^3+b^3+c^3 + 24abc}}$$

It says that this comes from the Jensen's inequality for $f(x) = \frac{1}{\sqrt{x}}$, but I couldn't figoure out how. Is this inequality true? You can prove without Jensen's inequality. If it's not, can you give a counterexample?

Answer 1

It's true also by Holder because $$\sum_{cyc}\frac{a}{\sqrt{a^2+8bc}}=\sqrt{\frac{\left(\sum\limits_{cyc}\frac{a}{\sqrt{a^2+8bc}}\right)^2\sum\limits_{cyc}a(a^2+8bc)}{\sum\limits_{cyc}a(a^2+8bc)}}\geq$$ $$\geq\sqrt{\frac{(a+b+c)^3}{\sum\limits_{cyc}a(a^2+8bc)}}=\frac{1}{\sqrt{a^3+b^3+c^3+24abc}}.$$