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.

Let B be the unit ball in C n with respect to an arbitrary norm and let f (z, t) be a g-Loewner chain such that e −t f (z, t) − z has a zero of order k + 1 at z = 0. In this paper, we obtain growth and covering theorems for f (·, 0). Moreover, we consider coefficient bounds and examples of mappings in S 0 g,k+1 (B).

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a b s t r a c tWe give a distortion theorem for linearly invariant families on the unit ball B of a finite dimensional JB * -triple X by using the trace-order. The exponents in the distortion bounds depend on the Bergman metric at 0. Further, we introduce a new definition for the traceorder of a linearly invariant family on B, based on a Jacobian argument. We also construct an example of a linearly invariant family on B which has minimum trace-order and is not a subset of the normalized convex mappings of B for dim X ≥ 2. 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. 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.

Let B be the unit ball in a complex Banach space. Let S * k+1 (B) be the family of normalized starlike mappings f on B such that z = 0 is a zero of order k + 1 of f (z) − z.The authors obtain sharp growth and covering theorems, as well as sharp coefficient bounds for various subsets of S * k+1 (B).