Note on the Fekete--Szeg\"{o} problem for spirallike mappings in Banach spaces

TL;DR

This paper clarifies a relationship between existing theorems on spirallike mappings in complex spaces and derives a sharp Fekete--Szeg"{o} estimate for such mappings in Banach spaces, simplifying proofs and extending results.

Contribution

It shows that one theorem in prior work is a consequence of another, and extends a sharp Fekete--Szeg"{o} estimate to general Banach spaces.

Findings

One theorem is a direct consequence of another in finite-dimensional spaces.

A sharp Fekete--Szeg"{o} estimate is established for Banach space mappings.

Simplifies the proof structure for coefficient inequalities in spirallike mappings.

Abstract

In this note we present a remark on the paper "On the coefficient inequalities for a class of holomorphic mappings associated with spirallike mappings in several complex variables" by Y.~Lai and Q.~Xu \cite{LX} published recently in the journal {\it Results in Mathematics}. We show that one of the theorems in \cite{LX} concerning the finite-dimensional space $\mathbb{C}^n$ is a direct consequence of another one, so it does not need an independent proof. Moreover, we prove that a sharp norm estimate on the Fekete--Szeg\"{o} functional over spirallike mappings in a general Banach space can be deduced from a result in \cite{LX}.