F. Wiener’s Trick and an Extremal Problem for $$H^p$$

Abstract

AbstractFor $$0<p \le \infty $$ 0 < p ≤ ∞ , let $$H^p$$ H p denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the kth Taylor coefficient of a function $$f \in H^p$$ f ∈ H p which satisfies $$\Vert f\Vert _{H^p}\le 1$$ ‖ f ‖ H p ≤ 1 and $$f(0)=t$$ f ( 0 ) = t for some $$0 \le t \le 1$$ 0 ≤ t ≤ 1 . In particular, we provide a complete solution to this problem for $$k=1$$ k = 1 and $$0<p<1$$ 0 < p < 1 . We also study F. Wiener’s trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.