Quasiconformal deformations of nonvanishing $H^p$ functions and the Hummel-Scheinberg-Zalcman conjecture

TL;DR

This paper extends the proof of the Hummel-Scheinberg-Zalcman conjecture to all Hardy spaces $H^p$ with $p \\ge 2$, confirming coefficient bounds for nonvanishing functions.

Contribution

The paper generalizes previous results by proving the conjecture for all $H^p$ spaces with $p \\ge 2$, completing the proof for these function spaces.

Findings

Confirmed the conjecture for all $H^p$ with $p \\ge 2$

Established coefficient bounds for nonvanishing $H^p$ functions

Connected results to the Krzyz conjecture for bounded functions

Abstract

Recently the author proved that the 1977 Hummel-Scheinberg-Zalcman conjecture on coefficients of nonvanishing $H^p$ functions is true for all $p = 2m, m \in \mathbb{N}$, i.e., for the Hilbertian Hardy spaces $H^{2m}$. As a consequence, this also implies a proof of the Krzyz conjecture for bounded nonvanishing functions which originated this direction. In the present paper, we solve the problem for all spaces $H^p$ with $p \ge 2$.