A lower bound for the $L_2[-1,\,1]$-norm of the logarithmic derivative of polynomials with zeros on the unit circle

Let $C$ be the unit circle $\{z:|z|=1\}$ and $Q_n(z)$ be an arbitrary $C$-polynomial (i.e., all its zeros $z_1,\dots, z_n\in C$). We prove that the norm of the logarithmic derivative $Q_n'/Q_n$ in the complex space $L_2[-1, 1]$ is greater than $1/8$.