Korean J. Math. Vol. 29 No. 2 (2021) pp.293-304
DOI: https://doi.org/10.11568/kjm.2021.29.2.293

Some remarks on the Subordination Principle for analytic functions concerned with Rogosinski's lemma

Main Article Content

Tuğba Akyel


In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for
bounded analytic functions. For new inequalities, the results of Rogosinski's lemma, Subordination principle and Jack's lemma were used.

Article Details


[1] T. Akyel and B. N. Ornek, Some Remarks on Schwarz lemma at the boundary, Filomat, 31 (13) (2017), 4139-4151. Google Scholar

[2] T. A. Azero˘glu and B. N. Ornek, ¨ A refined Schwarz inequality on the boundary, Complex Variables and Elliptic Equations 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 disc, 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. Mateljevic, N. Mutavdzc and BN. Ornek, ¨ Note on Some Classes Google Scholar

[9] of Holomorphic Functions Related to Jack’s and Schwarz’s Lemma, DOI: Google Scholar

[10] 13140/RG.2.2.25744.15369, ResearchGate. Google Scholar

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

[12] P. R. Mercer, An improved Schwarz Lemma at the boundary, Open Mathematics, 16 (2018), 1140-1144. 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. Ornek and T. D¨uzenli, Bound Estimates for the Derivative of Driving Point Impedance Functions, Filomat, 32(18) (2018), 6211-6218. Google Scholar

[15] B. N. Ornek and T. D¨uzenli, 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. Ornek, ¨ Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50(6) (2013), 2053-2059. Google Scholar

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