Analytic functions which are regular in a disc and smooth on its boundary

A theorem is established, asserting that the norm of the derivative $f^{(n)}(z)$ in the space $H^2$ for a function $f(z)$ regular in the disc is not increased if we replace $f$ by the ratio $f(z)/G(z)$, where $G(z)$ is any interior function dividing $f(z)$ whose singular part is of a particular form.