Bohr’s theorem for holomorphic mappings with values in homogeneous balls

Abstract

Let X, Y be complex Banach spaces. Let G be a bounded balanced domain in X and B Y be the unit ball in Y . Assume that B Y is homogeneous. Let f : G → B Y be a holomorphic mapping. In this paper, we show that, ifsuch that ϕ P (P ) = 0. Moreover, we show that the constant 1/3 is best possible, if B Y is the unit ball of a J * -algebra. The above result was proved by Liu and Wang in the case that G = B Y is one of the four classical domains in the sense of Hua. This result generalises a classical result of Bohr.

Some generalizations of Bohr's theorem

“…In this paper, we will generalize Bohr's theorem to holomorphic mappings f : G → B Y , where G is a bounded balanced domain in a complex Banach space X , and B Y is the unit ball in a JB*‐triple Y . Our result is a slight generalization of a result in .…”

“…Then, Kaup showed in ,Corollary 3.6. In , the following lemma was proved for the unit ball of a J*‐algebra. We generalize the result in to the unit ball of a JB*‐triple.…”

“…On the other hand, using homogeneous expansions of holomorphic functions, Aizenberg obtained another generalization of Bohr's theorem in ,Theorem 8; Liu and Wang gave a generalization of Bohr's theorem to holomorphic mappings of B into itself, where B is one of the four classical domains in in . Hamada, Honda and Kohr generalized to infinite dimensional spaces in . We refer for other generalizations of Bohr's theorem to and .…”

Trace-order and a distortion theorem for linearly invariant families on the unit ball of a finite dimensional JB∗ -triple

“…Finally, we prove a regularity theorem for linearly invariant families on B. All four types of classical Cartan domains are the open unit balls of JB * -triples, and the same holds for any finite product of these domains [22]; see also [23,24]. Thus the unit balls of JB * -triples are natural generalizations of the unit disc in C and we have a setting in which a large number of bounded symmetric homogeneous domains may be studied simultaneously.…”

On the Bohr Inequality

“…Earlier results on the generalization of Bohr theorem using homogeneous expansions can also be found in [12,Theorem 8] and [14]. Meanwhile, the generalization of both the results [58] and [12,Theorem 8] was obtained by Hamada et al [49].…”