How to prove this

[Nany]

Question:

Let A =(1+R)〖(-4+R^2)〗^2

B =〖(6+4R+B^2 R-5R^2-R^3+R^4+B(R^2+R-2))〗^2

C =(1+γ)〖(-4+γ^2)〗^2)

D =〖(6+4γ+β^2 γ-5γ^2-γ^3+γ^4+β(γ^2+γ-2))〗^2 )

I need to prove that A /4B - C /9D > 0, given 0 < γ < R < 1, 0 < β < B < 1.

 

How to do it? If I can use Mathematica to help, that will be great too. What I am thinking now is to find the Minimum of (A /4B - C /9D). If the minimum >0, then A /4B - C /9D >0. Is that a right direction to go? How can I write the input code for this including the constraints 0 < γ < R < 1, 0 < β < B < 1?

 

Thank you so much for any hints or help!