A Class of Interpolating Blaschke Products and Best Approximation in L ^p for p < 1

Abstract

We describe the inner functions Θ such that

$$\mid 1+\Theta f \mid_{H^p}^p \ge 1-\mid \Theta(0)\mid^2$$

for all p > 0 and f ∈ H ^p. We prove that each such inner function Θ satisfying Θ (0) ≠ 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 ∈ H ^p for which 1 + Θf does not vanish in the unit disk.