An extremal property of outer functions

Our main result is the following: if $f(z)$ is in the space $H^2$, and $F(z)$ is its outer part, then $\|F^{(n)}\|_{H^2}\le\|f^{(n)}\|_{H^2}$ $(n=1,2,\dots)$, the left side being finite if the right side is finite. Under certain essential restrictions, this inequality was proved by B.I. Korenblyum and V.S. Korolevich [1].