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http://dx.doi.org/10.7468/jksmeb.2016.23.1.61

AN IMPROVED LOWER BOUND FOR SCHWARZ LEMMA AT THE BOUNDARY  

ORNEK, BULENT NAFI (DEPARTMENT OF COMPUTER ENGINEERING, AMASYA UNIVERSITY)
AKYEL, TUGBA (DEPARTMENT OF MATHEMATICS, GEBZE TECHNICAL UNIVERSITY)
Publication Information
The Pure and Applied Mathematics / v.23, no.1, 2016 , pp. 61-72 More about this Journal
Abstract
In this paper, a boundary version of the Schwarz lemma for the holom- rophic function satisfying f(a) = b, |a| < 1, b ∈ ℂ and ℜf(z) > α, 0 ≤ α < |b| for |z| < 1 is invetigated. Also, we estimate a modulus of the angular derivative of f(z) function at the boundary point c with ℜf(c) = a. The sharpness of these inequalities is also proved.
Keywords
angular derivative; holomorphic function; Schwarz lemma on the boundary;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 H.P. Boas: Julius and Julia: Mastering the Art of the Schwarz lemma. Amer. Math. Monthly 117 (2010), 770-785.   DOI
2 D. Chelst: A generalized Schwarz lemma at the boundary. Proc. Amer. Math. Soc. 129 (2001), 3275-3278.   DOI
3 V.N. Dubinin: The Schwarz inequality on the boundary for functions regular in the disc. J. Math. Sci. 122 (2004), 3623-3629.   DOI
4 V.N. Dubinin: Bounded holomorphic functions covering no concentric circles. J. Math. Sci. 207 (2015), 825-831.   DOI
5 G.M. Golusin: Geometric Theory of Functions of Complex Variable [in Russian]. 2nd edn., Moscow 1966.
6 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.
7 M. Jeong: The Schwarz lemma and boundary fixed points. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (2011), 219-227.
8 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
9 X. Tang & T. Liu: The Schwarz lemma at the boundary of the egg domain Bp1,p2 in ℂn. Canad. Math. Bull. 58 (2015), 381-392.   DOI
10 X. Tang, T. Liu & J. Lu: Schwarz lemma at the boundary of the unit polydisk in ℂn. Sci. China Math. 58 (2015), 1-14.
11 M. Mateljević: The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings. Filomat 29:2 (2015), 221-244.   DOI
12 M. Mateljević: Distortion of harmonic functions and harmonic quasiconformal quasi-isometry. Revue Roum. Math. Pures Appl. 51 (2006) 56, 711-722.
13 M. Mateljević: Ahlfors-Schwarz lemma and curvature. Kragujevac J. Math. 25 (2003), 155-164.
14 M. Mateljević: Note on rigidity of holomorphic mappings & Schwarz and Jack lemma (in preparation). ResearchGate.
15 Ch. Pommerenke: Boundary behaviour of conformal maps. Springer-Verlag, Berlin, 1992.
16 R. Osserman: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128 (2000), 3513-3517.   DOI
17 T. Aliyev Azeroğlu & B.N. Örnek: A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations 58 (2013), 571-577.   DOI
18 B.N. Örnek: Sharpened forms of the Schwarz lemma on the boundary. Bull. Korean Math. Soc. 50 (2013), 2053-2059.   DOI
19 M. Elin, F. Jacobzon, M. Levenshtein & D. Shoikhet: 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