Suppose $R(z)$ is a rational function such that $|R(z)|=1$ for $|z|=1$. Let ${\alpha_j}^N_{j=1}$ be zeros and poles of $R(z)$ of order $m_j$ in the unit disc $|z|\lt 1$. Here $m_j\gt 0$ is $\alpha_j$ is a zero and $m_j\lt 0$ is it is a pole. Define $B(z)=[(z-\alpha_1)/(1-z\alpha'1)]^{m_1}...[(z-\alpha_N)/(1-z\alpha'_N)]^{m_N}$ (Where $\alpha'$ stands for the complex conjugate of $\alpha$.) Then how to show that $R(z)/B(z)$ is a bounded entire function? And how to show that $R(z)=\lambda B(z)$ for some $\lambda$ belongs to complex number with $|\lambda|=1$.
I've proved that $\alpha$ is a zero or a pole of order m iff $1/\alpha'$ is a pole or zero of order m respectively, but don't know how to proceed from there. This problem seems pretty difficult. Any help?