Prove that If $a,b,c \in R^+$ Then

$$\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{a+c}}+\sqrt{\frac{c}{a+b}} \gt 2$$

My try:

I assumed:




Then we have:


Now by AM GM Inequality we have:

$$\left(\frac{a}{b}+\frac{b}{a}\right) \ge 2$$

So we get:

$$\frac{1}{x^2}+\frac{1}{y^2}+\frac{1}{z^2} \ge 6 \tag{1}$$

Using $(1)$ we need to prove:

$$x+y+z \gt 2$$

Any idea here?

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  • $\begingroup$ (1) does not give you $x+y+z >2$. $\endgroup$ – Kavi Rama Murthy Jan 31 '19 at 9:12
  • $\begingroup$ but how can $c=0$, $a,b,c$ are positive reals $\endgroup$ – Umesh shankar Jan 31 '19 at 9:17
  • $\begingroup$ @Umeshshankar Right, you should probably mention that $a, b, c > 0$, because $\mathbb{R}^+$ is a bit ambiguous whether it includes $0$. $\endgroup$ – orlp Jan 31 '19 at 9:18

Writing cyclic sums for short, $$\sum_{cyc}\sqrt{\frac{a}{b+c}}=\sum_{cyc}\frac{2a}{2\sqrt{a(b+c)}}\\ \geq\sum_{cyc}\frac{2a}{a+b+c}=2 $$ where the AM-GM inequality was used for each denominator, i.e. $2\sqrt{a(b+c)} \le a + (b+c)$

As Martin R. pointed out, math.stackexchange.com/a/2237066/42969 has the same reasoning, so credits belong to Michael Rozenberg. Here, some extra explanation is given.

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  • $\begingroup$ So is the inequality $\ge 2$? $\endgroup$ – Umesh shankar Jan 31 '19 at 9:30
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    $\begingroup$ Yes. To be more precise, equality would only hold in the AM-GM terms if $a = b+c$, $b = a+c$, $c = b+a$ which would entail $a=b=c=0$ which is not possible. So we can exclude this case, and $>$ holds. $\endgroup$ – Andreas Jan 31 '19 at 9:33
  • $\begingroup$ So you mean to say when AM GM is used there are some exceptions like this problems, where equality cannot occur. $\endgroup$ – Umesh shankar Jan 31 '19 at 9:39
  • $\begingroup$ Well, what do you mean by "exceptions"? I was explaning that equality in AM-GM - as used here - holds if and only if the two terms are equal (in all three cases where AM-GM was used). Then I argued that if $a,b,c \in R^+$, this equality condition cannot occur. So I concluded that indeed $>$ holds. $\endgroup$ – Andreas Jan 31 '19 at 9:45
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    $\begingroup$ Yes that's how I would put it. $\endgroup$ – Andreas Jan 31 '19 at 10:31

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