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Asymptotically spirallike mappings in several complex variables

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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(B n) 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) = 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(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* = 2I n , then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings.

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Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Ian Graham

  2. Faculty of Engineering, Kyushu Sangyo University, 31 Matsukadai 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan

    Hidetaka Hamada

  3. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 1 M. Kogălniceanu Str., 400084, Cluj-Napoca, Romania

    Gabriela Kohr & Mirela Kohr

Authors
  1. Ian Graham
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  2. Hidetaka Hamada
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  3. Gabriela Kohr
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  4. Mirela Kohr
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Correspondence to Ian Graham.

Additional information

Partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221.

Partially supported by Grant-in-Aid for Scientific Research (C) no. 19540205 from Japan Society for the Promotion of Science, 2007.

Partially supported by Romanian Ministry of Education and Research, CEEX Program, Project 2-CEx06-11-10/2006.

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Graham, I., Hamada, H., Kohr, G. et al. Asymptotically spirallike mappings in several complex variables. J Anal Math 105, 267–302 (2008). https://doi.org/10.1007/s11854-008-0037-1

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