The rational maps $z \mapsto 1 + 1/\omega z^d$ have no Herman rings

We prove that for every $d \in \mathbb N, d \geq 2$, the rational maps in the family $\{ z \mapsto 1 + 1/\omega z^d : \omega \in \textbf{C} \setminus \{0 \} \}$ have no Herman rings. From this we conclude a dynamical characterization for the parameters in the Mandelbrot set of these families. Further, we show that hyperbolic maps are dense in this family if and only if the set of parameters for which the Julia set is the whole sphere has no interior.