Puzzle: Double smaller-too

Definitions:
arithmetic mean A=(a+b)/2
geometric mean G=sqrt(ab)
harmonic mean H=(2ab)/(a+b)
a>b>0

It is well known inequality A > G > H

It is easy to show A-H > G-H, but here is hopefully funny puzzle:

Proof or disproof : A-H > 2(G-H)

Enjoy! :-)

Tapio


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