Computational Methods and Function Theory 2 (2002), No. 2, 549--578
Copyright Heldermann Verlag 2002
Alexei B. Aleksandrov
alex@pdmi.ras.ru, St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka, 27, 191023, St. Petersburg, Russia.
We describe the inner functions $\Theta$ such that $$ \|1+\Theta f\|_{H^p}^p \ge 1-|\Theta(0)|^2 $$ for all $p>0$ and $f\in H^p$. We prove that each such inner function $\Theta$ satisfying $\Theta(0)\not=0$ is an interpolating Blaschke product. Moreover, we study the inner functions such that $\|1+\Theta f\|_{H^p}^p\ge1-|\Theta(0)|^2$ for all $p>0$ and for all $f\in H^p$ for which $1+\Theta f$ does not vanish in the unit disk.
Keywords: Hardy spaces, Blaschke products, best approximation.
MSC 2000: 30D50, 30D55.
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