<s>[INST] Prove that $\frac{1}{1 - \sqrt{ab}} + \frac{1}{1 - \sqrt{bc}} + \frac{1}{1 - \sqrt{ca}} \leq \frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c}$ Given that $0 < a , b , c < 1$.
Prove that $\frac{1}{1 -  \sqrt{ab}} + \frac{1}{1 - \sqrt{bc}} + \frac{1}{1 - \sqrt{ca}} \leq \frac{1}{1 -  a} + \frac{1}{1 - b} + \frac{1}{1 - c}$.
I tried using modified C.S. and brute-force. But , it demands a lot of calculation. So , I want some better solution than this. Thank you.
 [/INST] We have $\sqrt{ab}\leq \frac{a+b}{2}$ so $1-\sqrt{ab}\geq 1-\frac{a+b}{2}>0$, which implies
$$\dfrac{1}{1-\sqrt{ab}}\leq \dfrac{2}{2-a-b}$$
And using the simple inequality $\frac{4}{x+y}\leq \frac{1}{x}+\frac{1}{y}$ for all $x,y>0$ you get
$$\dfrac{1}{1-\sqrt{ab}}\leq \dfrac{2}{2-a-b}\leq \dfrac{1}{2(1-a)}+\dfrac{1}{2(1-b)}$$
Summing up gives the result.
</s>