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Solutions for the generalized Loewner differential equation in several complex variables

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Abstract

We generalize a one-variable result of J. Becker to several complex variables. We determine the form of arbitrary solutions of the Loewner differential equation that is satisfied by univalent subordination chains of the form \({f(z, t)=e^{tA}z+\cdots,}\) where \({A\in L(\mathbb{C}^n, \mathbb{C}^n)}\) has the property k +(A) < 2m(A). Here \({k_+(A)=\max\{\Re\lambda:\lambda\in \sigma(A)\}}\) and \({m(A)=\min\{\Re\langle A(z), z \rangle: \|z\|=1\}}\) . (The notion of parametric representation has a useful generalization under these conditions, so that one has a canonical solution of the Loewner differential equation.) In particular, we determine the form of the univalent solutions. The results are applied to subordination chains generated by spirallike mappings on the unit ball in \({\mathbb{C}^n}\) . Finally, we determine the form of the solutions in the presence of certain coefficient bounds.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    Peter Duren

  2. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Ian Graham

  3. Faculty of Engineering, Kyushu Sangyo University, 3-1 Matsukadai 2-Chome, Higashi-ku, Fukuoka, 813-8503, Japan

    Hidetaka Hamada

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

    Gabriela Kohr

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

Additional information

I. Graham is partially supported by the Natural Sciences and Engineering Research Council of Canada under Grant A9221. G. Kohr is partially supported by the UEFISCSU-CNCSIS Grants PN-II-ID 524/2007 and 525/2007.

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Duren, P., Graham, I., Hamada, H. et al. Solutions for the generalized Loewner differential equation in several complex variables. Math. Ann. 347, 411–435 (2010). https://doi.org/10.1007/s00208-009-0429-2

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