F. Wiener's trick and an extremal problem for $H^p$

TL;DR

This paper solves an extremal problem for the first Taylor coefficient in Hardy spaces for 0<p<1, utilizing Wiener's trick to analyze coefficient bounds under norm and value constraints.

Contribution

It provides a complete solution for the extremal problem when k=1 and 0<p<1, and explores the application of Wiener's trick in Hardy space coefficient problems.

Findings

Explicit extremal functions for k=1, 0<p<1

Wiener's trick effectively bounds coefficients in Hardy spaces

Enhanced understanding of coefficient extremal problems

Abstract

For $0<p \leq \infty$, let $H^p$ denote the classical Hardy space of the unit disc. We consider the extremal problem of maximizing the modulus of the $k$th Taylor coefficient of a function $f \in H^p$ which satisfies $\|f\|_{H^p}\leq1$ and $f(0)=t$ for some $0 \leq t \leq 1$. 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.