No, there are no such functions with $c>1$. Your condition means that they expand the spherical metric. If you look at the area of the whole sphere, you obtain a contradiction.
More precisely, looking at the area of the sphere and the area of its image we conclude that the area of the image is finite, therefore your function is rational (and $c$ is its degree). But a rational function of degree $>1$ must have a critical point the derivative is 0 at this point, which is in contradiction with your assumption.