Suppose $\phi$ is an invertible function in $L^\infty(T)$ where $T$ is the unit circle such that the essential range of $\phi$, $R(\phi)$ is contained in the open right half-plane. Show there exist an $\epsilon>0$ such that $\epsilon R(\phi)\subset \{z\in\mathbb{C}:|z-1|<1\}$.
This is part of a proof. I cannot see why it is true. I know $R(\phi)$ is compact and does not contain 0 as $\phi$ is invertible.
For this, we have $R(\phi)=\{\lambda\in\mathbb{C}:\mu(\{x\in X:|f(x)-\lambda|<\epsilon\})>0 \ \forall \epsilon>0\}$. Here $\mu$ is the Lebesgue measure and $X$ is the unit circle.