Browse > Article
http://dx.doi.org/10.7468/jksmeb.2017.24.3.129

BEHAVIOR OF HOLOMORPHIC FUNCTIONS ON THE BOUNDARY OF THE UNIT DISC  

Ornek, Bulent Nafi (Department of Computer Engineering, Amasya University)
Publication Information
The Pure and Applied Mathematics / v.24, no.3, 2017 , pp. 129-145 More about this Journal
Abstract
In this paper, we establish lower estimates for the modulus of the non-tangential derivative of the holomorphic functionf(z) at the boundary of the unit disc. Also, we shall give an estimate below |f''(b)| according to the first nonzero Taylor coefficient of about two zeros, namely z = 0 and $z_0{\neq}0$.
Keywords
Schwarz lemma on the boundary; holomorphic function; second non-tangential derivative; Jack's lemma;
Citations & Related Records
Times Cited By KSCI : 2  (Citation Analysis)
연도 인용수 순위
1 T. Aliyev Azeroglu & B.N. Ornek: A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations 58 (2013), 571-577.   DOI
2 H.P. Boas: Julius and Julia: Mastering the Art of the Schwarz lemma. Amer. Math. Monthly 117 (2010), 770-785.   DOI
3 D. Chelst: A generalized Schwarz lemma at the boundary. Proc. Amer. Math. Soc. 129 (2001), 3275-3278.   DOI
4 V.N. Dubinin: The Schwarz inequality on the boundary for functions regular in the disc. J. Math. Sci. 122 (2004), 3623-3629.   DOI
5 V.N. Dubinin: Bounded holomorphic functions covering no concentric circles J. Math. Sci. 207 (2015), 825-831.   DOI
6 G.M. Golusin: Geometric Theory of Functions of Complex Variable [in Russian]. 2nd edn., Moscow 1966.
7 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.
8 M. Jeong: The Schwarz lemma and boundary fixed points. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (2011), 219-227.
9 D.M. Burns & S.G. Krantz: Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary. J. Amer. Math. Soc. 7 (1994), 661-676.   DOI
10 T. Liu & X. Tang: The Schwarz Lemma at the Boundary of the Egg Domain $B_{p1,p2}$ in ${\mathbb{C}}^n$. Canad. Math. Bull. 58 (2015), 381-392.   DOI
11 T. Liu, X. Tang & J. Lu: Schwarz lemma at the boundary of the unit polydisk in ${\mathbb{C}}^n$. Sci. hina Math. 58 (2015), 1-14.
12 M. Mateljevic: Schwarz lemma, the Carath-eodory and Kobayashi Metrics and Appli-cations in Complex Analysis. XIX GEOMETRICAL SEMINAR, At Zlatibor, Sunday, August 28, 2016 Sunday, September 4, 2016.
13 M. Mateljevic: Distortion of harmonic functions and harmonic quasiconformal quasi-isometry. Revue Roum. Math. Pures Appl. 51 (2006), 711-722.
14 B.N. Ornek: Sharpened forms of the Schwarz lemma on the boundary. Bull. Korean Math. Soc. 50 (2013), 2053-2059.   DOI
15 M. Mateljevic: Ahlfors-Schwarz lemma and curvature. Kragujevac J. Math. 25 (2003), 155-164.
16 M. Mateljevic: Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation). ResearchGate.
17 R. Osserman: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128 (2000), 3513-3517.   DOI
18 Ch. Pommerenke: Boundary Behaviour of Conformal Maps. Springer-Verlag, Berlin, 1992.
19 D. Shoikhet, M. Elin, F. Jacobzon & M. Levenshtein: The Schwarz lemma: Rigidity and Dynamics, Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.
20 H. Unkelbach: Uber die Randverzerrung bei konformer Abbildung. Math. Z., 43 (1938), 739-742.   DOI