Can This Algebraic Inequality Be Proven Using AM-GM and GM-HM Methods?

[Albert1]

a,b,c >0 , prove that :

$(1+\dfrac {a}{b})(1+\dfrac {b}{c})(1+\dfrac {c}{a})\geq 2(1+\dfrac {a+b+c}{\sqrt[3]{abc}})$

 

[Albert1]

Re: Prove an inequality

a,b,c >0 , prove that :
$(1+\dfrac {a}{b})(1+\dfrac {b}{c})(1+\dfrac {c}{a})\geq 2(1+\dfrac {a+b+c}{\sqrt[3]{abc}})$

using the AM-GM inequality for left side
expansion of left side =$2+k=2+\dfrac{k}{3}+\dfrac {2k}{3}\geq 4+\dfrac {2k}{3}$
where $k=\dfrac {b}{a} +\dfrac{a}{b}+\dfrac{c}{b} +\dfrac{b}{c}+\dfrac{a}{c} +\dfrac{c}{a}\geq 6 $
using the GM-HM inequality for right side
right side :$\leq 2(1+\dfrac {(a+b+c)}{\dfrac {3}{(1/a) + (1/b) + (1/c)}} =
2(1+\dfrac {(a+b+c)\times \left [(1/a) + (1/b) + (1/c)\right ]}{3} )=4+\dfrac {2k}{3}$
$\therefore $ the proof is done