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

Abstract

For 0<p≤∞, let Hp 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∈Hp which satisfies ∥f∥Hp≤1 and f(0)=t for some 0≤t≤1. In particular, we provide a complete solution to this problem for k=1 and 0<p<1. We also study F. Wiener’s trick, which plays a crucial role in various coefficient-related extremal problems for Hardy spaces.