Some estimates for angular derivative at the boundary

Bülent Nafi Örnek

Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 3 (2017), pp. 120-134.

Résumé

In this paper, we establish lower estimates for the modulus of the values of $f(z)$ on boundary of unit disc. For the function $f(z)=1+c_1z+c_2z^2+\dots$ defined in the unit disc such that $f(z)\in\mathcal N(\beta)$ assuming the existence of angular limit at the boundary point $b$, the estimations below of the modulus of angular derivative have been obtained at the boundary point $b$ with $f(b)=\beta$. Moreover, Schwarz lemma for class $\mathcal N(\beta)$ is given. The sharpness of these inequalities has been proved.

Bibliographie

[1] Aliyev Azeroğlu T., Örnek B. N., “A refined Schwarz inequality on the boundary”, Complex Variables and Elliptic Equations, 58 (2013), 571–577 | DOI | MR | Zbl

[2] Boas H. P., “Julius and Julia: Mastering the Art of the Schwarz lemma”, Amer. Math. Monthly, 117 (2010), 770–785 | DOI | MR | Zbl

[3] Chelst D., “A generalized Schwarz lemma at the boundary”, Proc. Amer. Math. Soc., 129 (2001), 3275–3278 | DOI | MR | Zbl

[4] Dubinin V. N., “The Schwarz inequality on the boundary for functions regular in the disc”, J. Math. Sci., 122 (2004), 3623–3629 | DOI | MR | Zbl

[5] Dubinin V. N., “Bounded holomorphic functions covering no concentric circles”, J. Math. Sci., 207 (2015), 825–831 | DOI | MR | Zbl

[6] Golusin G. M., Geometric Theory of Functions of Complex Variable, 2nd edn., Moscow, 1966 (in Russian) | MR

[7] Jeong M., “The Schwarz lemma and its applications at a boundary point”, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 21 (2014), 275–284 | MR

[8] Jeong M., “The Schwarz lemma and boundary fixed points”, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math., 18 (2011), 219–227 | MR

[9] Krantz S. G., Burns D. M., “Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary”, J. Amer. Math. Soc., 7 (1994), 661–676 | DOI | MR | Zbl

[10] Liu T., Tang X., “The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1,p_2}$ in $\mathbb C^n$”, Canad. Math. Bull., 58 (2015), 381–392 | DOI | MR | Zbl

[11] Lu J., Tang X., Liu T., “Schwarz lemma at the boundary of the unit polydisk in $\mathbb C^n$”, Sci. China Math., 58 (2015), 1–14 | MR

[12] Mateljević M., “Schwarz lemma, the Carathéodory and Kobayashi Metrics and Applications in Complex Analysis”, XIX Geometrical Seminar, At Zlatibor, 2016, 1–12

[13] Mateljević M., Hyperbolic geometry and Schwarz lemma, ResearchGate, 2016

[14] Mateljević M., “Ahlfors–Schwarz lemma and curvature”, Kragujevac J. Math., 25 (2003), 155–164 | MR | Zbl

[15] Mateljević M., Note on Rigidity of Holomorphic Mappings Schwarz and Jack Lemma, ResearchGate, (in preparation)

[16] Osserman R., “A sharp Schwarz inequality on the boundary”, Proc. Amer. Math. Soc., 128 (2000), 3513–3517 | DOI | MR | Zbl

[17] Örnek B. N., “Sharpened forms of the Schwarz lemma on the boundary”, Bull. Korean Math. Soc., 50 (2013), 2053–2059 | DOI | MR | Zbl

[18] Pommerenke Ch., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992 | MR | Zbl

[19] Shoikhet D., Elin M., Jacobzon F., Levenshtein M., “The Schwarz lemma: Rigidity and Dynamics”, Harmonic and Complex Analysis and its Applications, Springer International Publishing, 2014, 135–230 | MR | Zbl

[20] Unkelbach H., “Uber die Randverzerrung bei konformer Abbildung”, Math. Z., 43:1 (1938), 739–742 | DOI | MR | Zbl