Korean J. Math. Vol. 31 No. 1 (2023) pp.25-33
DOI: https://doi.org/10.11568/kjm.2023.31.1.25

Some results for the class of analytic functions concerned with symmetric points

Article Sidebar

PDF
Published: Mar 30, 2023
Keywords:
Analytic function, Schwarz lemma, Julia-Wolff lemma, Angular derivative.
Mathematics Subject Classification:
30C80

Main Article Content

Ayse Nur Arabaci
Department of Mathematics, Amasya University, Merkez-Amasya, 05100, Turkey.
Bülent Nafi Örnek
Department of Computer Engineering, Amasya University, Merkez-Amasya, 05100, Turkey.

Abstract

This paper's objectives are to present the $\mathcal{H}$ class of analytical functions and explore the many characteristics of the functions that belong to this class. Some inequalities regarding the angular derivative have been discovered for the functions in this class. In addition, the symmetry points on the unit disc are used for the obtained inequalities.


Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

References

[1] T. Akyel and B. N. Ornek, Sharpened forms of the Generalized Schwarz inequality on the boundary, Proc. Indian Acad. Sci. (Math. Sci.), 126 (1) (2016), 69–78. Google Scholar

[2] T. A. Azero ̆glu and B. N. O ̈rnek, A refined Schwarz inequality on the boundary, Complex Variab. Elliptic Equa. 58 (2013), 571–577. Google Scholar

[3] H. P. Boas, Julius and Julia: Mastering the Art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), 770–785. Google Scholar

[4] V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disk, J. Math. Sci. 122 (2004), 3623–3629. Google Scholar

[5] G. M. Golusin, Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Moscow 1966. Google Scholar

[6] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc. 3 (1971), 469–474. Google Scholar

[7] M. Mateljevi ́c, Rigidity of holomorphic mappings & Schwarz and Jack lemma, DOI:10.13140/RG.2.2.34140.90249, In press. Google Scholar

[8] M. Mateljevi ́c, N. Mutavdˇz ́c and B. N. O ̈rnek, Note on some classes of holomorphic functions related to Jack’s and Schwarz’s lemma, Appl. Anal. Discrete Math., 16 (2022), 111–131. Google Scholar

[9] P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski’s lemma, Journal of Classical Analysis 12 (2018), 93–97. Google Scholar

[10] P. R. Mercer, An improved Schwarz Lemma at the boundary, Open Mathematics 16 (2018), 1140–1144. Google Scholar

[11] M. Nunokawa, J. Sok ́ol and H. Tang, An application of Jack-Fukui-Sakaguchi lemma, Journal of Applie Analysis and Computation, 10 (2020), 25–31. Google Scholar

[12] M. Nunokawa and J. Sok ́ol, On a boundary property of analytic functions, J. Ineq. Appl., 2017:298 (2017), 1–7. Google Scholar

[13] R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517. Google Scholar

[14] B. N. O ̈rnek and T. Akyel, On bounds for the derivative of analytic functions at the boundary, Korean J. Math., 29 (4) (2021), 785–800. Google Scholar

[15] B. N. O ̈rnek and T. Du ̈zenli, Boundary Analysis for the Derivative of Driving Point Impedance Functions, IEEE Transactions on Circuits and Systems II: Express Briefs 65 (9) (2018) 1149–1153. Google Scholar

[16] B. N. O ̈rnek and T. Du ̈zenli, On Boundary Analysis for Derivative of Driving Point Impedance Functions and Its Circuit Applications, IET Circuits, Systems and Devices, 13 (2) (2019), 145–152. Google Scholar

[17] Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin. 1992. Google Scholar

[18] H. Unkelbach, U ̈ber die Randverzerrung bei konformer Abbildung, Math. Z., 43 (1938), 739–742. Google Scholar