Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Applications of the Schwarz Lemma and Jack's Lemma for the Holomorphic Functions

  • Received : 2018.07.26
  • Accepted : 2019.04.23
  • Published : 2020.09.30
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Abstract

We consider a boundary version of the Schwarz Lemma on a certain class of functions which is denoted by 𝒩. For the function f(z) = z + a2z2 + a3z3 + … which is defined in the unit disc D such that the function f(z) belongs to the class 𝒩, we estimate from below the modulus of the angular derivative of the function ${\frac{f{^{\prime}^{\prime}}(z)}{f(z)}}$ at the boundary point c with f'(c) = 0. The sharpness of these inequalities is also proved.

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References

  1. T. A. Azeroglu and B. N. Ornek, A refined Schwarz inequality on the boundary, Complex Var. Elliptic Equ., 58(2013), 571-577. https://doi.org/10.1080/17476933.2012.718338
  2. H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly, 117(2010), 770-785. https://doi.org/10.4169/000298910x521643
  3. D. M Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary J. Amer. Math. Soc., 7(1994), 661-676. https://doi.org/10.1090/S0894-0347-1994-1242454-2
  4. D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc., 129(2001), 3275-3278. https://doi.org/10.1090/S0002-9939-01-06144-5
  5. V. N. Dubinin, On the Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122(2004), 3623-3629. https://doi.org/10.1023/B:JOTH.0000035237.43977.39
  6. G. M. Golusin, Geometric theory of functions of complex variable, Translations of Mathematical Monographs 26, American Mathematical Society, Providence, R.I., 1969.
  7. I. S. Jack, Functions starlike and convex of order ${\alpha}$, J. London Math. Soc., 3(1971), 469-474. https://doi.org/10.1112/jlms/s2-3.3.469
  8. M. Jeong, The Schwarz lemma and its application at a boundary point, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 21(2014), 219-227.
  9. M. Mateljevic, Hyperbolic geometry and Schwarz lemma, Symposium MATHEMATICS AND APPLICATIONS, Vol. VI(1), Faculty of Mathematics, University of Belgrade, 2015.
  10. M. Mateljevic, Schwarz lemma, the Caratheodory and Kobayashi metrics and applications in cmplex analysis, XIX GEOMETRICAL SEMINAR, At Zlatibor., (2016), 1-12.
  11. M. Mateljevic, Rigidity of holomorphic mappings, Schwarz and Jack lemma, DOI:10.13140/RG.2.2.34140.90249.
  12. P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. Appl., 205(1997), 508-511. https://doi.org/10.1006/jmaa.1997.5217
  13. P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski's lemma, J. Class. Anal., 12(2018), 93-97. https://doi.org/10.7153/jca-2018-12-08
  14. R. Singh and S. Singh, Some sufficient conditions for univalence and starlikeness, Colloq. Math., 47(1982), 309-314. https://doi.org/10.4064/cm-47-2-309-314
  15. B. N. Ornek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc., 50(2013), 2053-2059. https://doi.org/10.4134/BKMS.2013.50.6.2053
  16. B. N. Ornek, Inequalities for the non-tangential derivative at the boundary for holomorphic function, Commun. Korean Math. Soc., 29(2014), 439-449. https://doi.org/10.4134/CKMS.2014.29.3.439
  17. B. N. Ornek, Inequalities for the angular derivatives of certain classes of holomorphic functions in the unit disc, Bull. Korean Math. Soc., 53(2016), 325-334. https://doi.org/10.4134/BKMS.2016.53.2.325
  18. B. N. Ornek, Estimates for holomorphic functions concerned with Jack's lemma, Publ. Inst. Math., 104(118)(2018), 231-240. https://doi.org/10.2298/PIM1818231O
  19. R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc., 128(2000), 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0
  20. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.