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Univalence Criteria and Quasiconformal Extensions

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Ukrainian Mathematical Journal Aims and scope

We establish more general conditions for the univalence of analytic functions in the open unit disk \( \mathcal{U} \). In addition, we obtain a refinement of the criterion of quasiconformal extension for the main result.

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References

  1. L. V. Ahlfors, “Sufficient conditions for quasiconformal extension,” Ann. Math. Stud., 79, 23–29 (1974).

    MathSciNet  Google Scholar 

  2. J. M. Anderson and A. Hinkkanen, “Univalence criteria and quasiconformal extensions,” Trans. Amer. Math. Soc., 324, 823–842 (1991).

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. J. Becker, “Conformal mappings with quasiconformal extensions,” in: Asp. Contemp. Complex Anal., D. A. Brannan and J. G. Clunie (Eds.), Academic Press, New York (1980), pp. 37–77.

  6. Th. Betker, “Löewner chains and quasiconformal extensions,” Complex Var. Theory Appl., 20, No. 1–4, 107–111 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Çağlar and H. Orhan, “Some generalizations on the univalence of an integral operator and quasiconformal extensions,” Miskolc Math. Notes, 14, No. 1, 49–62 (2013).

  8. E. Deniz, “Sufficient conditions for univalence and quasiconformal extensions of meromorphic functions,” Georgian Math. J., DOI 10.1515/gmj-2012-0027, 15 p.

  9. E. Deniz, D. Raducanu, and H. Orhan, “On an improvement of a univalence criterion,” Math. Balkanica (N.S.), 24, No. 1-2, 33–39 (2010).

  10. E. Deniz and H. Orhan, “Some notes on extensions of basic univalence criteria,” J. Korean Math. Soc., 48, No. 1, 179–189 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  11. M. Goluzin, “Geometric theory of functions of a complex variable,” Amer. Math. Soc. Trans. Math. Monogr., Providence, RI, 29 (1969).

  12. I. Hotta, “Löewner chains with complex leading coefficient,” Monatsh. Math., 163, 315–325 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. Hotta, “Explicit quasiconformal extensions and Loewner chains,” Proc. Japan Acad. Ser. A Math. Sci., 85, 108–111 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  14. I. Hotta, Löewner Chains and Quasiconformal Extension of Univalent Functions, Dissertation, Tohoku Univ. (2010).

  15. S. Kanas and H. M. Srivastava, “Some criteria for univalence related to Ruscheweyh and Salagean derivatives,” Complex Var. Elliptic Equ., 38, 263–275 (1997).

    Article  MATH  Google Scholar 

  16. J. G. Krýz, “Convolution and quasiconformal extension,” Comment. Math. Helv., 51, 99–104 (1976).

  17. J. Miazga and A. Wesolowski, “A univalence criterion and the structure of some subclasses of univalent functions,” Ann. Univ. Mariae Curie-Skłodowska Sect. A, 40, No. 16, 153–161 (1986).

    MathSciNet  MATH  Google Scholar 

  18. Z. Nehari, “The Schwarzian derivate and schlicht functions,” Bull. Amer. Math. Soc. (N.S.), 55, 545–551 (1949).

  19. S. Ozaki and M. Nunokawa, “The Schwarzian derivative and univalent functions,” Proc. Amer. Math. Soc., 33, No. 2, 392–394 (1972).

    Article  MathSciNet  MATH  Google Scholar 

  20. H. Ovesea-Tudor and S. Owa, “An extension of the univalence criteria of Nehari and Ozaki,” Hokkaido Math. J., 34, No. 3, 533–539 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  21. N. N. Pascu and V. Pescar, “A generalization of Pfaltzgraff’s theorem,” Sem. Geometr. Funct. Theory, 2, 91–98 (1991).

    MathSciNet  MATH  Google Scholar 

  22. J. A. Pfaltzgraff, “k-Quasiconformal extension criteria in the disk,” Complex Var., 21, 293–301 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  23. Ch. Pommerenke, “Univalent function,” in: Vandenhoech Ruprecht in Göttingen (1975).

  24. D. Pommerenke, “On a univalence criterion,” Mathematica, 37(60), No. 1-2, 227–231 (1995).

  25. D. Raducanu, H. Orhan, and E. Deniz, “On some sufficient conditions for univalence,” An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat., 18, No. 2, 217–222 (2010).

  26. D. Raducanu, I. Radomir, M. E. Gageonea, and N. R. Pascu, “A generalization of Ozaki–Nunokawa’s univalence criterion,” J. Inequal. Pure Appl. Math., 5, No. 4, Article 95 (2004).

  27. D. Raducanu, “A univalence criterion for analytic functions in the unit disk,” Mathematica, 46(69), No. 2, 213–216 (2004).

  28. D. Raducanu and H. Tudor, “A generalization of Goluzin’s univalence criterion,” Stud. Univ. Babeş-Bolyai Math., 57, 261–267 (2012).

  29. H. Tudor, “New univalence criteria,” Stud. Univ. Babeş-Bolyai. Math., 52, No. 2, 127–132 (2007).

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 8, pp. 1147–1152, August, 2016.

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Çağlar, M., Orhan, H. Univalence Criteria and Quasiconformal Extensions. Ukr Math J 68, 1314–1321 (2017). https://doi.org/10.1007/s11253-017-1297-7

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  • DOI: https://doi.org/10.1007/s11253-017-1297-7

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