Admissibility of majorants in certain model subspaces: necessary conditions

A nonnegative function $\omega$ on $\mathbb{R}$ is called an admissible majorant for an inner function $\Theta$ if there is a nonzero function $f\in H^2\ominus\Theta H^2$ such that $|f|\le\omega$. Some conditions necessary for admissibility are presented in the case where $\Theta$ is meromorphic.