For positive reals $a,\,b,\,c$, prove that [MATH]\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\gt 2[/MATH].

Hello readers!

In my previous blog post, I asked the readers to spot the factual mistake(s) that I might have or might not have made in the solution (of mine) to one delicious inequality problem.

Today, I am going to discuss with you the mistake that I intentionally made.

From the following steps that I made:

[MATH]\frac{a}{\sqrt{a}\sqrt{b+c}}+\frac{b}{\sqrt{b}\sqrt{c+a}}+\frac{c}{\sqrt{c}\sqrt{a+b}}[/MATH]

[MATH]\gt \frac{a}{\frac{a+b+c}{2}}+\frac{b}{\frac{a+b+c}{2}}+\frac{c}{\frac{a+b+c}{2}}[/MATH] (from the AM-GM inequality)

I used the AM-GM inequalities below to get to the next step:

[MATH]\frac{a+(b+c)}{2}\ge \sqrt{a}\sqrt{b+c}[/MATH], with equality when $a=b+c$,

[MATH]\frac{b+(c+a)}{2}\ge \sqrt{b}\sqrt{c+a}[/MATH], with equality when $b=c+a$,

[MATH]\frac{c+(a+b)}{2}\ge \sqrt{c}\sqrt{a+b}[/MATH], with equality when $c=a+b$.

[MATH]\frac{a}{\frac{a+b+c}{2}}+\frac{b}{\frac{a+b+c}{2}}+\frac{c}{\frac{a+b+c}{2}}[/MATH]

[MATH]\gt 2\left(\frac{a}{a+b+c}+\frac{b}{a+b+c}+\frac{c}{a+b+c}\right)[/MATH]

[MATH]= 2\left(\frac{a+b+c}{a+b+c}\right)[/MATH]

[MATH]= 2 [/MATH](Q.E.D.)

But to attain the equality, the conditions $a=b+c$, $b=c+a$, and $c=a+b$ must be satisfied. In other words, we got to have $a+b+c=0$. But we're told all $a,\,b$ and $c$ are positive reals, so their sum can't equal to zero.

This means we must omit the equality sign and thus, we will end up with the strict inequality:

 [MATH]\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\gt 2[/MATH]

i.e. for positive reals $a,\,b,\,c$, the minimum for [MATH]\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}[/MATH] isn't a $2$.