How to Find an Analytic Function Given Specific Conditions?

[sparkster]

So let f be analytic in the open unit disk and continuous on the closed unit disk. Also, |f(z)|=1 for |z|=1, all zeros are simple zeros at 0, and f'(0)=-1/2.

I need to find f.

I've tried using the cauchy integral formula for f' but that's not getting me anywhere. Can anyone point me in the right direction?

 

[ObsessiveMathsFreak]

Perhaps you could use the fact that the real an imaginary parts of a complex differentiable function are harmonic, i.e.
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2} = \frac{\partial^2 v}{\partial x^2}+\frac{\partial^2 v}{\partial y^2}= 0$$
in the disc.

Also on the boundary
$$|f|=1=|f|^2=u^2+v^2=1$$

And for the derivative you have
$$f'(0)=\frac{\partial u}{\partial x} + i \frac{\partial v}{\partial x}=\frac{\partial v}{\partial y} - i \frac{\partial u}{\partial y}=-\frac{1}{2}$$

This gives
$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}=-\frac{1}{2}$$
$$\frac{\partial v}{\partial x}=\frac{\partial u}{\partial y}=0$$
At the origin.

But is this enough to solve the partial differential equation?