Asymptotically spirallike mappings in several complex variables

Abstract

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space ℂn. We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(ℂn, ℂn), and prove that if \( \int_0^\infty {\parallel e^{(A - 2m(A)I_n )t} \parallel } \)dt < ∞ (this integral is convergent if k+(A) < 2m(A)), then a mapping f ∈ S(Bn) is A-asymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f(z, t) such that D f(0, t) = eAt, {e−Atf(·, t)}t≥0 is a normal family on Bn and f = f(·, 0). In particular, a spirallike mapping with respect to A ∈ L(ℂn, ℂn) with \( \int_0^\infty {\parallel e^{(A - 2m(A)I_n )t} \parallel } \)dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A* = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.