Inequalites of triangle side with $abc = 1$

Question

Let $a,b,c$ be the sides of a triangle with $abc=1$. Prove that $$ \frac{\sqrt{b+c−a}}{a} + \frac{\sqrt{c+a-b}}{b} + \frac{\sqrt{a+b−c}}{c} \ge a+b+c $$

Answer 1

By Holder $\left(\sum\limits_{cyc}\frac{\sqrt{b+c-a}}{a}\right)^2\sum\limits_{cyc}a^2(b+c-a)^2\geq(a+b+c)^3$.

Hence, it remains to prove that $abc(a+b+c)\geq\sum\limits_{cyc}a^2(b+c-a)^2$, which is

$\sum\limits_{cyc}(a-b)^2(a+c-b)(b+c-a)\geq0$, which is obvious.