A note on differentials of holomorphic functions

Recently, in [Aron R, Dimant V, García-Lirola LC, et al. Linearization of holomorphic Lipschitz functions. Math Nachr. 2024;297(8):3024–3051. doi:10.1002/mana.v297.8], a bridge was made between the very active area of spaces of Lipschitz real functions on a metric space and holomorphic functions on an open subset of a Banach space. This was done by introducing and studying the space HL0(BX) of holomorphic Lipschitz functions defined on BX, the open unit ball of the complex Banach space X vanishing at 0. It was proved there that this space is isometrically isomorphic to a subspace of H∞(BX,X∗), the space of bounded holomorphic mapping with values in the topological dual of X. In that paper, it was shown that this subspace was proper, except in the one-dimensional case. The goal of this note is to give an intrinsic characterization of the elements of that subspace. Moreover, in the case where X additionally has a Schauder basis, it is shown that there is an explicit way to calculate whether an element of H∞(BX,X∗) belongs or not to that subspace.