Admissibility criteria for model subspaces with fast growth of the argument of the generating inner function

Let $\Theta$ be an inner function in the upper half plane and let $K_\Theta=H^2\ominus\Theta H^2$ be the associated model subspace of the Hardy space $H^2$. We call a non-negative function $\omega$ $\Theta$-admissible if in the space $K_\Theta$ there exists a non-zero function $f\in K_\Theta$ such that $|f|\leq\omega$ a.e. on $\mathbb{R}$. We give some sufficient conditions of $\Theta$-admissibility for the case when $\Theta$ is meromorphic and $\arg\Theta$ grows fast ($(\arg\Theta)'$ tends to infinity).