Show that a function analytic in the disc $\left | z \right |<1+\epsilon$ for some $\epsilon>0$ satisfies $\displaystyle f(z)=i Im(f(0))+\frac{1}{2\pi}\int_0^{2\pi}\frac{e^{it}+z}{e^{it}-z}Re(f(e^{it}))dt$
Suppose that the function above also satisfies $Re(f(e^{it}))\geq0, \forall t\in[0,2\pi]$, prove that: $\displaystyle \frac{1-\left | z \right |}{1+\left | z \right |}Re(f(0))\leq Re(f(0))\leq\frac{1+\left | z \right |}{1-\left | z \right |}Re(f(0)),\forall z: \left | z \right |<1$
I don't even know where to begin ._.