Help Me Prove this Identity (or find a counterexample)

[Poopsilon]

Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a way to prove it:

$$\overline{f(0)} = \frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i\phi}}{e^{i \phi}-z}\overline{f(e^{i\phi})}d\phi$$

The functions I tried were: z, z+1, and z+i, so nothing transcendental. We don't know that $\overline{f(z)}$ is analytic and we can't even push the conjugate inside the function, thus I feel like I don't have many tools at my disposal except for algebraic manipulation, and so far that hasn't gotten me anywhere.

 

[Dustinsfl]

Let f be an analytic function defined in an open set containing the closed unit disk and let z in ℂ be fixed. I've simplified a more complicated expression down to this identity, and as implausible as it looks, after some numerical checking it does in fact appear to be true, but I can't find a way to prove it:

$$\overline{f(0)} = \frac{1}{2\pi}\int_0^{2\pi}\frac{e^{i\phi}}{e^{i \phi}-z}\overline{f(e^{i\phi})}d\phi$$

The functions I tried were: z, z+1, and z+i, so nothing transcendental. We don't know that $\overline{f(z)}$ is analytic and we can't even push the conjugate inside the function, thus I feel like I don't have many tools at my disposal except for algebraic manipulation, and so far that hasn't gotten me anywhere.

What is the original question?