  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}.
