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Löwner chains with complex leading coefficient

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Abstract

In this paper we confirm that several crucial theorems due to Pommerenke and Becker for the theory of Löwner chains work well without normalization on the complex-valued first coefficient. As applications of those considerations, some new univalent and quasiconformal extension criteria are given in the last section.

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References

  1. Becker J.: Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen. J. Reine Angew. Math. 255, 23–43 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Becker J.: Über die Lösungsstruktur einer Differentialgleichung in der konformen Abbildung. J. Reine Angew. Math. 285, 66–74 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Becker J.: Conformal Mappings with Quasiconformal Extensions. Aspects of Contemporary Complex Analysis, pp. 37–77. Academic Press, London (1980)

    Google Scholar 

  4. Betker Th.: Löwner chains and quasiconformal extensions. Complex Var. Theory Appl. 20(1–4), 107–111 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Brown J.E.: Quasiconformal extensions for some geometric subclasses of univalent functions. Int. J. Math. Math. Sci. 7(1), 187–195 (1984)

    Article  MATH  Google Scholar 

  6. Conway, J.B.: Functions of One Complex Variable. II, Graduate Texts in Mathematics, vol. 159. Springer, New York (1995)

  7. Gall, U.: Über das Randverhalten von Bazilevič-Funktionen. Dissertation an der Technischen Universität Berlin (1986)

  8. Graham, I., Kohr, G.: Geometric function theory in one and higher dimensions. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 255. New York (2003)

  9. Hotta, I.: Ruscheweyh’s univalent criterion and quasiconformal extensions. Kodai Math. J. (to appear)

  10. Hotta I.: Explicit quasiconformal extensions and Löwner chains. Proc. Jpn. Acad. Ser. A Math. Sci. 85(8), 108–111 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Krzyż J.G.: Quasiconformal extensions of some special univalent functions. Colloq. Math. 51, 189–193 (1987)

    MathSciNet  MATH  Google Scholar 

  12. Miller, S.S., Mocanu, P.T.: Differential subordinations, theory and applications. In: Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. New York (2000)

  13. Pommerenke Ch.: On starlike and convex functions. J. Lond. Math. Soc. 37, 209–224 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  14. Pommerenke Ch.: Über die Subordination analytischer Funktionen. J. Reine Angew. Math. 218, 159–173 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pommerenke Ch.: Univalent Functions. Vandenhoeck & Ruprecht, Göttingen (1975)

    MATH  Google Scholar 

  16. Ruscheweyh S.: An extension of Becker’s univalence condition. Math. Ann. 220(3), 285–290 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sheil-Small T.: On Bazilevič functions. Quart. J. Math. Oxford Ser. 23(2), 135–142 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Division of Mathematics, Graduate School of Information Sciences, Tohoku University, 6-3-09 Aramaki-Aza-Aoba, Aoba-ku, Sendai, 980-8579, Japan

    Ikkei Hotta

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  1. Ikkei Hotta
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Correspondence to Ikkei Hotta.

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Hotta, I. Löwner chains with complex leading coefficient. Monatsh Math 163, 315–325 (2011). https://doi.org/10.1007/s00605-010-0200-5

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  • DOI: https://doi.org/10.1007/s00605-010-0200-5

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