In resticted domain , Applying the Cauchy-Schwarz's inequality

I see this problem a few days ago.

$$a,b,c\in [\alpha ,\beta ]$$

prove that

$$9\le (a+b+c){\textstyle (}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}{\textstyle )}\le \frac{(2\alpha +\beta )(\alpha +2\beta )}{\alpha \beta}$$

Left inequality is easy by Cauchy-Schwart's inequality. But Right inequality is difficult to me. How can I approach the Right inequlity?

EDIT : Sorry, I forgot the condition : $\alpha ,\beta >0$

I see this problem a few days ago.

$$a,b,c\in [\alpha ,\beta ]$$

prove that

$$9\le (a+b+c){\textstyle (}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}{\textstyle )}\le \frac{(2\alpha +\beta )(\alpha +2\beta )}{\alpha \beta}$$

Left inequality is easy by Cauchy-Schwart's inequality. But Right inequality is difficult to me. How can I approach the Right inequlity?

EDIT : Sorry, I forgot the condition : $\alpha ,\beta >0$