(Sqrt[α + γ (((-1 + ϵ))^2) + (-2 + ϵ) ϵ]) + (Sqrt[α + γ (((1 + ϵ))^2) - ϵ (2 + ϵ)]) < 
      2 (Sqrt[α + γ])

S1 = (Sqrt[α + γ (((-1 + ϵ))^2) + (-2 + ϵ) ϵ]);
S2 = (Sqrt[α + γ (((1 + ϵ))^2) - ϵ (2 + ϵ)]);
S3 = 2 (Sqrt[α + γ]);
Expand[(S1 + S2)^2]  ≥  S3^2

Simplify[%]

LHS = (Sqrt[α + γ (((-1 + ϵ))^2) + (-2 + ϵ) ϵ]) (Sqrt[α + γ (((1 + ϵ))^2) - ϵ (2 + ϵ)]);
RHS = α + γ + 
          2 ϵ - γ (ϵ^2);
Expand[LHS^2 ≥ RHS^2]

Simplify[%]

NotebookPrint[
      SelectedNotebook[], "/home/fdahl/writing/lis/notebooks/sqrt \
inequality.nb.ps"]