Hi, having trouble with a few complex analysis problems. Here they are with my thoughts - I'd appreciate any help you can offer - thanks.

1) Find all conformal mappings from the upper half plane to itself which fix the points at zero and infinity (easy to see this should be a linear transformation, but the constant is supposedly in the range $\displaystyle (1, \infty )$, which I'm not seeing.)

2) Let $\displaystyle f(x)=x^3+ax^2+bx+c$ with a,b,c real, a positive, and c negative. How many zeroes does $f(x)$ have in the left half plane? (Tried using Rouche but couldn't really get a conclusive answer.)

3) Find a harmonic function on the intersection of a disk and the upper half plane (forgot the boundary conditions, etc. Just looking for a general map to a wedge-shaped region, the rest is elementary.)

Thanks a ton.