A refinement of Schwarz's lemma at the boundary

  • Bülent Nafi Örnek Department of Computer Engineering, Amasya University, Turkey
Keywords: Analytic function, Angular derivative, Schwarz lemma


UDC 517.5

We investigate a boundary version of the Schwarz lemma for analytic functions.  In addition, an analytic function satisfying the equality case is found by deducing inequalities related to the modulus of the derivative of analytic functions at a boundary point of the unit disk.  Some coefficients used in the Taylor expansion of the function are considered in these inequalities.  In the last theorem, by analyzing the Taylor expansion about two points, we obtain the modulus of the derivative of the function at point 1.


T. Aliyev Azeroğlu, B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Var. and Elliptic Equat., 58, 571–577 (2013).

H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly, 117, 770–785 (2010).

V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122, 3623–3629 (2004).

G. M. Golusin, Geometric theory of functions of complex variable} (in Russian), 2nd ed., Moscow (1966).

M. Mateljević, N. Mutavdžć, B. N. Örnek, Note on some classes of holomorphic functions related to Jack's and Schwarz's lemma, Appl. Anal. and Discrete Math., 16, 111–131 (2022).

P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski's lemma, J. Class. Anal., 12, 93–97 (2018).

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

B. N. Örnek, Estimates for analytic functions connected with Hankel determinant, Ukr. Math. J., 73, 1398–1411 (2022).

B. N. Örnek, T. Düzenli, Boundary analysis for the derivative of driving point impedance functions, IEEE Trans. Circuits and Syst. II. Express Briefs, 65, № 9, 1149–1153 (2018).

B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc., 50, 2053–2059 (2013).

B. N. Örnek, S. B. Aydemir, T. Düzenli, B. Özak, Some remarks on activation function design in complex extreme learning using Schwarz lemma, Neurocomputing, 492, 23–33 (2022).

Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin (1992).

H. Unkelbach, Über die Randverzerrung bei konformer Abbildung, Math. Z., 43, 739–742 (1938).

How to Cite
Örnek, B. N. “A Refinement of Schwarz’s Lemma at the Boundary”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 76, no. 4, Apr. 2024, pp. 515 -24, doi:10.3842/umzh.v74i4.7364.
Research articles