Inequality $\sum_{cyc} \sqrt{\frac{a}{b+c+d}}>2$

Question

Let $a,b,c,d>0$. I want to prove that $$\sum_{cyc} \sqrt{\frac{a}{b+c+d}}>2.$$

I try using Hölder's inequality : $$\left(\sum_{cyc} b+c+d\right)\left(\sum_{cyc} \sqrt{\frac{a}{b+c+d}}\right)^2\geq\left(\sum_{cyc} \sqrt[3] a\right)^3$$

so it would be enough to prove $$(\sqrt[3] a+ \sqrt[3] b+ \sqrt[3] c+ \sqrt[3] d)^3\geq 12(a+b+c+d)$$

Answer 1

Your last inequality is wrong, try for example $a=b=\frac1{100}$ and $c=d=\frac{1}{1000}$. However, we can proceed as follows:

Lemma. For $a,b,c,d>0$, we have $$\sqrt{\frac{a}{b+c+d}}\geq \frac{2a}{a+b+c+d}.$$

Proof. Note that $$(a-b-c-d)^2\geq0$$ so that $$(a + b + c + d)^2-4 a (b + c + d)\geq 0$$ which means that $$(a + b + c + d)^2\geq 4a (b + c + d).$$

Hence, we have $$\frac{a}{b+c+d}\geq\frac{4a^2}{(a+b+c+d)^2},$$ which proves the Lemma. $\square$

We can now conclude since $$\sum_{\text{cyc}} \sqrt{\frac{a}{b+c+d}}\geq \sum_{\text{cyc}} \frac{2a}{a+b+c+d}=2.$$

Equality occurs only if $a=b=c=d=0$ which is not possible.