## Subordination chains and the growth theorem of spirallike mappings.(English)Zbl 1027.46094

From the introduction: Let $$f$$ be a univalent mapping in the unit disc $$\Delta$$ with $$f(0)=0$$ and $$f'(0)=1$$. Then the classical growth theorem is as follows: $\frac{|z|}{(1+|z|)^2}\leq|f(z)|\leq\frac{|z|}{(1-|z|)^2}.$ In this paper, we generalize the above growth theorem to spirallike mappings of type $$\alpha$$ on the unit ball B in an arbitrary complex Banach space. One might think that the same growth theorem holds for all normalized spirallike mappings defined by T. J. Suffridge [in: Adv. Compl. Funct. Theory, Proc. Semin. Maryland Univ. 1973/74, Lect. Notes Math. 505, 164-203 (1976; Zbl 0324.30018)]. However, we can give an example of a normalized spirallike mapping such that the same growth theorem does not hold. This example also shows that the growth of normalized spirallike mappings cannot be estimated from above.
We also give an alternate characterization of normalized spirallike mappings of type $$\alpha$$ on the unit ball B with respect to an arbitrary norm on $${\mathbb C}^n$$ in terms of subordination chains.

### MSC:

 46T25 Holomorphic maps in nonlinear functional analysis 32A30 Other generalizations of function theory of one complex variable 30C55 General theory of univalent and multivalent functions of one complex variable

Zbl 0324.30018