Research Article
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Some remarks for a certain class of holomorphic functions at the boundary of the unit disc

Year 2019, Volume: 23 Issue: 3, 446 - 452, 01.06.2019
https://doi.org/10.16984/saufenbilder.464294

Abstract

We consider a boundary version of the Schwarz Lemma on a certain class which is
denoted by K(alfa) . For the function  f(z)=z+c2z2+....... which is defined in the unit disc E
such that the function f (z) belongs to the class K(alfa) , we estimate from below the modulus of the
angular derivative of the function ( )
( )
zf ' z
f z
at the boundary point b with ( ) = 1
( ) 1
bf ' b
f b 
. Moreover,
we get the Schwarz Lemma for the class K(alfa) . We also investigate some inequalities obtained in
terms of sharpness.

References

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  • D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), 3275-3278.
  • V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. 122 (2004), 3623-3629.
  • V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207 (2015), 825-831.
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  • I. S. Jack, Functions starlike and convex of order alfa , J. London Math. Soc. 3(1971), 469-474.
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  • X. Tang and T. Liu, The Schwarz Lemma at the Boundary of the Egg Domain p1, p2 B in Cn , Canad. Math. Bull. 58 (2015), 381-392.
  • X. Tang, T. Liu and J. Lu, Schwarz lemma at the boundary of the unit polydisk in Cn , Sci. China Math. 58 (2015), 1-14.
  • M. Mateljević, Rigidity of holomorphic mappings & Schwarz and Jack lemma, DOI: 10.13140/RG.2.2.34140.90249, In press.
  • R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517.
  • T. Aliyev Azeroğlu and B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Variables and Elliptic Equations 58 (2013), 571–577.
  • B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), 2053–2059.
  • B. N. Örnek, Estimate for p-valently Functions at the Boundary, Bulletin of International Mathematical Virtual Institute, 7 (2017), 327-338.
  • Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
  • M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet, The Schwarz lemma: Rigidity and Dynamics, Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.
Year 2019, Volume: 23 Issue: 3, 446 - 452, 01.06.2019
https://doi.org/10.16984/saufenbilder.464294

Abstract

References

  • H. P. Boas, Julius and Julia: Mastering the Art of the Schwarz lemma, Amer. Math. Monthly, 117 (2010), 770-785.
  • D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), 3275-3278.
  • V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. 122 (2004), 3623-3629.
  • V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207 (2015), 825-831.
  • G. M. Golusin, Geometric Theory of Functions of Complex Variable [inRussian], 2nd edn., Moscow 1966.
  • I. S. Jack, Functions starlike and convex of order alfa , J. London Math. Soc. 3(1971), 469-474.
  • M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), 275-284.
  • 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.
  • X. Tang and T. Liu, The Schwarz Lemma at the Boundary of the Egg Domain p1, p2 B in Cn , Canad. Math. Bull. 58 (2015), 381-392.
  • X. Tang, T. Liu and J. Lu, Schwarz lemma at the boundary of the unit polydisk in Cn , Sci. China Math. 58 (2015), 1-14.
  • M. Mateljević, Rigidity of holomorphic mappings & Schwarz and Jack lemma, DOI: 10.13140/RG.2.2.34140.90249, In press.
  • R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517.
  • T. Aliyev Azeroğlu and B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Variables and Elliptic Equations 58 (2013), 571–577.
  • B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), 2053–2059.
  • B. N. Örnek, Estimate for p-valently Functions at the Boundary, Bulletin of International Mathematical Virtual Institute, 7 (2017), 327-338.
  • Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
  • M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet, The Schwarz lemma: Rigidity and Dynamics, Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Articles
Authors

Bülent Nafi Örnek 0000-0001-7109-230X

Tuğba Akyel This is me 0000-0001-6484-2731

Publication Date June 1, 2019
Submission Date September 26, 2018
Acceptance Date January 21, 2019
Published in Issue Year 2019 Volume: 23 Issue: 3

Cite

APA Örnek, B. N., & Akyel, T. (2019). Some remarks for a certain class of holomorphic functions at the boundary of the unit disc. Sakarya University Journal of Science, 23(3), 446-452. https://doi.org/10.16984/saufenbilder.464294
AMA Örnek BN, Akyel T. Some remarks for a certain class of holomorphic functions at the boundary of the unit disc. SAUJS. June 2019;23(3):446-452. doi:10.16984/saufenbilder.464294
Chicago Örnek, Bülent Nafi, and Tuğba Akyel. “Some Remarks for a Certain Class of Holomorphic Functions at the Boundary of the Unit Disc”. Sakarya University Journal of Science 23, no. 3 (June 2019): 446-52. https://doi.org/10.16984/saufenbilder.464294.
EndNote Örnek BN, Akyel T (June 1, 2019) Some remarks for a certain class of holomorphic functions at the boundary of the unit disc. Sakarya University Journal of Science 23 3 446–452.
IEEE B. N. Örnek and T. Akyel, “Some remarks for a certain class of holomorphic functions at the boundary of the unit disc”, SAUJS, vol. 23, no. 3, pp. 446–452, 2019, doi: 10.16984/saufenbilder.464294.
ISNAD Örnek, Bülent Nafi - Akyel, Tuğba. “Some Remarks for a Certain Class of Holomorphic Functions at the Boundary of the Unit Disc”. Sakarya University Journal of Science 23/3 (June 2019), 446-452. https://doi.org/10.16984/saufenbilder.464294.
JAMA Örnek BN, Akyel T. Some remarks for a certain class of holomorphic functions at the boundary of the unit disc. SAUJS. 2019;23:446–452.
MLA Örnek, Bülent Nafi and Tuğba Akyel. “Some Remarks for a Certain Class of Holomorphic Functions at the Boundary of the Unit Disc”. Sakarya University Journal of Science, vol. 23, no. 3, 2019, pp. 446-52, doi:10.16984/saufenbilder.464294.
Vancouver Örnek BN, Akyel T. Some remarks for a certain class of holomorphic functions at the boundary of the unit disc. SAUJS. 2019;23(3):446-52.