In a fresh start of Mathematica this
Condition34=Reduce[{(a-c β)^2(6 b^2+4 b λ+λ^2)/(8 b^3)>3(a-c α)^2/(4 b),
a>c>0,1>p>0,1>t>0,1>b>0,0<λ<1,β>α>1,a>c α,a>c β},β]
instantly gives me
α<β<a/c-Sqrt[6]*Sqrt[(a^2*b^2-2*a*b^2*c*α+b^2*c^2*α^2)/(c^2*(6*b^2+4*b*λ+λ^2))]]
I have no evidence that it might be the cause of the problem, but you might remember that if you assign a value to a variable then Mathematica will silently remember that assignment until you clear that or restart Mathematica. Some users are surprised when hours or days later something doesn't work and it is finally discovered that they didn't realize a previous assigned value was being used.
Please check all this very carefully.
