Let's consider $h(z)$ analytic in $ B(0,1)$ and continuous in $\overline{B(0,1)}$ , such that $ Re(h(z))=0 $ in $\partial D(0,1)$. Prove that $h(z)$ is constant.
Well... Since $h$ is continuous on a compact set, then attains it's maximum and it's minimum on it. That points are in $\partial D(0,1)$ since otherwise , the function is constant ( maximum modulus principle). More than that I don't know what can I do )=