Let be $g\in L^\infty $. The Toeplitz operator $T_g$ on the Hardy space $H^2(\mathbb D)$ in the unit disc $\mathbb D$ is defined by $T_gf=P(gf)$ for all $f\in H^2(\mathbb D)$, where $P$ is the orthogonal projection from $L^2$ to $H^2(\mathbb D)$.
How can we show that $T_g$ is an expansion if the distance from 0 to the convex hull of the (essential) closure of $g(\mathbb T)$ is at least 1?
Here, expansion means that $T_g$ satisfy $||T_gf||\geq ||gf||$ for each $f\in H^2(\mathbb D)$.