Prove that $\sum_\text{cyc}\frac{a}{b^2}\ge 3\sum_{cyc}\frac{1}{a^2}$ for $a,b,c>0$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$

Question

$a$, $b$ and $c$ are three positives such that $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = 1$. Prove that $$\dfrac{a}{b^2} + \frac{b}{c^2} + \frac{c}{a^2} \ge 3 \cdot \left(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\right)$$

Here's what I did.

We have that

$$\left(\frac{a}{b^2} + \frac{b}{c^2} + \frac{c}{a^2}\right)\left(\frac{1}{b} + \frac{1}{c} + \frac{1}{a}\right) \ge \left(\sqrt{\frac{a}{b^3}} + \sqrt{\frac{b}{c^3}} + \sqrt{\frac{c}{a^3}}\right)^2$$

But because of $\dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} = 1$.

$$\implies \frac{a}{b^2} + \frac{b}{c^2} + \frac{c}{a^2} \ge 3 \cdot \left(\frac{1}{b}\sqrt{\frac{a}{c^3}} + \frac{1}{c}\sqrt{\frac{b}{a^2}} + \frac{1}{a}\sqrt{\frac{c}{b^2}}\right)$$

And I am stuck, I can't think anymore.

Answer 1

Let $x=\frac{1}{a}$ , $y=\frac{1}{b}$ and $z=\frac{1}{c}$, so $\sum_{cyc} x=1$. Then we have

$(LHS-RHS)\cdot xyz=(\sum_{cyc}x)\cdot(\sum_{cyc}y^3z)-3\sum_{cyc}x^3yz=\sum_{cyc}x^4y+\sum_{cyc}x^3y^2-2\sum_{cyc}x^3yz$ $=\frac{1}{7}\sum_{cyc}((7x^4y+4z^3x^2+2x^3y^2+y^3z^2)-14x^3yz)\geq0.$

The last inequality come from the GM-AM by viewing $7x^4y+4z^3x^2+2x^3y^2+y^3z^2$ as 14 terms.

Answer 2

Let $a=\min\{a,b,c\}$, $b=a+u$ and $c=a+v$.

Thus, $u\geq0$, $v\geq0$ and we need to prove that $$\sum_{cyc}\frac{1}{a}\sum_{cyc}\frac{a}{b^2}\geq3\sum_{cyc}\frac{1}{a^2}$$ or $$\sum_{cyc}(a^4c^3+a^4c^2b-2a^2b^3c)\geq0,$$ which is true because $$\sum_{cyc}(a^4c^3+a^4c^2b-2a^2b^3c)=4(u^2-uv+v^2)a^5+(5u^3-2u^2v+7uv^2+5v^3)a^4+$$ $$+2(u^4+3u^2v^2+6uv^3+v^4)a^3+uv(u^3+10uv^2+5v^3)a^2+2u^2v^3(u+2v)a+u^3v^3\geq0.$$