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http://dx.doi.org/10.7468/jksmeb.2019.26.2.59

ESTIMATES FOR A CERTAIN SUBCLASS OF HOLOMORPHIC FUNCTIONS  

Ornek, Bulent Nafi (Department of Computer Engineering, Amasya University)
Akyel, Tugba (Department of Computer Engineering, Maltepe University)
Publication Information
The Pure and Applied Mathematics / v.26, no.2, 2019 , pp. 59-73 More about this Journal
Abstract
In this paper, a version of the boundary Schwarz Lemma for the holomorphic function belonging to $\mathcal{N}$(${\alpha}$) is investigated. For the function $f(z)=z+c_2z^2+C_3z^3+{\cdots}$ which is defined in the unit disc where $f(z){\in}\mathcal{N}({\alpha})$, we estimate the modulus of the angular derivative of the function f(z) at the boundary point b with $f(b)={\frac{1}{b}}\int\limits_0^b$ f(t)dt. The sharpness of these inequalities is also proved.
Keywords
holomorphic function; Jack's lemma; angular derivative;
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Times Cited By KSCI : 1  (Citation Analysis)
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1 X. Tang & T. Liu: 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
2 X. Tang, T. Liu & J. Lu: Schwarz lemma at the boundary of the unit polydisk in ${\mathbb{C}}^n$. Sci. China Math. 58 (2015), 1-14.
3 M. Mateljevic: The Lower Bound for the Modulus of the Derivatives and Jacobian of Harmonic Injective Mappings. Filomat 29 (2015), no. 2, 221-244.   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 M. Mateljevic: Distortion of harmonic functions and harmonic quasiconformal quasi-isometry. Revue Roum. Math. Pures Appl. Vol. 51 (2006), 711-722.
6 M. Mateljevic: Ahlfors-Schwarz lemma and curvature. Kragujevac J. Math. 25 (2003), 155-164.
7 M. Mateljevic: Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation). ResearchGate.
8 R. Osserman: A sharp Schwarz inequality on the boundary. Proc. Amer. Math. Soc. 128 (2000), 3513-3517.   DOI
9 T. Aliyev Azeroglu & B.N. Ornek: A refined Schwarz inequality on the boundary. Complex Variables and Elliptic Equations 58 (2013), 571-577.   DOI
10 B.N. Ornek: Sharpened forms of the Schwarz lemma on the boundary. Bull. Korean Math. Soc. 50 (2013), 2053-2059.   DOI
11 Ch. Pommerenke: Boundary Behaviour of Conformal Maps. Springer-Verlag, Berlin, 1992.
12 M. Nunokawa & Y. Ota: Starlikeness of Libera transformation II. Kyoto University Research Information Repository 1062 (1998), 62-68.
13 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.
14 Taishun Liu: Jianfei Wang, Xiaomin Tang. Schwarz Lemma at the Boundary of the Unit Ball in ${\mathbb{C}}^n$ and Its Applications. J. Geom. Anal 25 (2015), 1890-1914.   DOI
15 H. Unkelbach: Uber die Randverzerrung bei konformer Abbildung. Math. Z. 43 (1938), 739-742.   DOI
16 G.M. Golusin: Geometric Theory of Functions of Complex Variable [in Russian]. 2nd edn., Moscow 1966.
17 H.P. Boas: Julius and Julia: Mastering the Art of the Schwarz lemma. Amer. Math. Monthly 117 (2010), 770-785.   DOI
18 D. Chelst: A generalized Schwarz lemma at the boundary. Proc. Amer. Math. Soc. 129 (2001), 3275-3278.   DOI
19 V.N. Dubinin: Bounded holomorphic functions covering no concentric circles. J. Math. Sci. 207 (2015), 825-831.   DOI
20 I.S. Jack: Functions starlike and convex of order ${\alpha}$. J. London Math. Soc. 3 (1971), 469-474.   DOI
21 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.
22 M. Jeong: The Schwarz lemma and boundary fixed points. J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (2011), 219-227.
23 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