A maximum principle for quotient norms in $H\sp \infty$

Let $G$ be a closed subset of the open unit disk $D$ in the complex plane, and let $p$ denote a general polynomial of degree $n$ which has all of its roots in $G$. For a fixed $h$ in ${H^\infty },\vert\vert h - p{H^\infty }\vert{\vert _{{H^\infty }/p{H^\infty }}}$ is maximized only if all the zeros of $p$ are on the boundary of $G$.