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 C n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric represen-, then a mapping f ∈ S(B n ) is Aasymptotically 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 Df (0, t) = e At , {e −At f (·, t)} t≥0 is a normal family on B n and f = f (·, 0). In particular, a spirallike mapping with respect to A ∈ L(C n , C n ) with ∞ 0 e (A−2m(A)In )t 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.

Convergence results for families of univalent mappings on the unit ball in C^n

“…The following growth result was obtained by Graham, Hamada, and Kohr [11] for the family M (see [13,Lemma 1.2] in the case of mappings h ∈ N ; see also [34,Proposition 1.2.3] for the family N ). [8]; cf.…”

“…In this case, the family S t A (B n ) reduces to the family S 0 A (B n ) of mappings with A-parametric representation on B n , for all t ≥ 0 (see [13]). If…”

“…It is easily seen that if A ∈ L(C n ) and f ∈ H(B n ) is a normalized mapping, then f ∈ S A (B n ) if and only if f (z, t) = e tA f (z) is a normal Loewner chain with respect to A (see [13]). …”

“…Since the early works devoted to Loewner chains and the Loewner differential equation in higher dimensions due to Pfaltzgraff [27] and Poreda [28,29], many results in this field have been obtained (see [1,5,6,9,11,13,14,15,20,21,35]). We also mention the main contributions of Bracci [5] related to the existence of bounded support points for the family S 0 (B n ), n ≥ 2, and of Roth [31] concerning the ndimensional version of the well known Pontryagin maximum principle.…”

“…We have considered examples and applications which yield that the study of the family S t A (B n ) for time-dependent operators A ∈ A is basically different from that in the case of constant time-dependent linear operators (see [22]). In the case that A(t) = A, for all t ≥ 0, where A ∈ L(C n ) with k + (A) < 2m(A), then S t A (B n ) = S 0 A (B n ), for all t ≥ 0, where S 0 A (B n ) is the family of mappings with A-parametric representation (see [13]). Note that k + (A) is the Lyapunov index of A and m(A) = min z =1 ℜ A(z), z .…”