Hi. You have two Sqrt functions. The larger one is the same concept as the smaller one, and easier for posting here...
For a Real solution, the function inside a Sqrt function should be >= 0. If it's negative, we switch to complex.
Sqrt[-s^2*(-a^2+a^2*y-y*b^2)];
equ = -s^2 (-a^2+a^2 y-b^2 y);
As you can see, the relationships may be more complex then just "assuming" a>0, etc.
Mathematica does not know which of the assumptions you need.
Reduce[equ>=0,{a,b,s,y},Reals]//FullSimplify
(a==0&&(b==0||s==0||y>=0))||(a!=0&&(Sqrt[a^2]==b||Sqrt[a^2]+b==0||s==0||(Abs[a]>b&&b+Abs[a]>0&&a^2/(a^2-b^2)>=y)||(a^2/(a^2-b^2)<=y&&(Abs[a]<b||b+Abs[a]<0))))
