# Coefficient estimates for $H^p$ spaces with $0<p<1$

## TL;DR

This paper calculates specific coefficient bounds for functions in the Hardy spaces with 0<p<1, providing explicit constants and identifying extremal functions.

## Contribution

It explicitly computes the constants C(2,p) for 0<p<1 and C(3,2/3), and characterizes the functions that attain these bounds.

## Key findings

- Computed C(2,p) for 0<p<1.
- Determined C(3,2/3).
- Identified extremal functions for these bounds.

## Abstract

Let $C(k,p)$ denote the smallest real number such that the estimate $|a_k|\leq C(k,p)\|f\|_{H^p}$ holds for every $f(z)=\sum_{n\geq0}a_n z^n$ in the $H^p$ space of the unit disc. We compute $C(2,p)$ for $0<p<1$ and $C(3,2/3)$, and identify the functions attaining equality in the estimate.