Invariant subspaces for Toeplitz operators

The article is devoted to the invariant subspace problem for Toeplitz operators.
Let $\Gamma$ be a Lipschitz arc on the complex plane, $f$ be a non-constant continuous function on the unit circle. If there exists an open disc $D$ such that $f(\mathbb T)\cap\Gamma\cap D\ne\varnothing$, $f(\mathbb T)\cap(\bar D\setminus\Gamma)\ne\varnothing$ and if the modulus of continuity $\omega_f$ of $f$ satisfies the inequality
$$ \int\limits_\bigcirc\frac{\omega_f(t)}{t\log\frac1t}\,dt<+\infty, $$
then non-trivial hyperinvariant subspaces for the Toeplitz operator $T_f$ on the Hardy class $H^2$ are proved to exist.
For the proof of this result the Lubich–Matsaev theorem is used.