Applications of the Jack's lemma for the meromorphic functions at the boundary
DOI:
https://doi.org/10.5269/bspm.v38i7.46633Abstract
In this paper, a boundary version of the Schwarz lemma for the class $\mathcal{% N(\alpha )}$ is investigated. For the function $f(z)=\frac{1}{z}% +a_{0}+a_{1}z+a_{2}z^{2}+...$ defined in the punctured disc $E$ such that $% f(z)\in \mathcal{N(\alpha )}$, we estimate a modulus of the angular derivative of the function $\frac{zf^{\prime }(z)}{f(z)}$ at the boundary point $c$ with $\frac{cf^{\prime }(c)}{f(c)}=\frac{1-2\beta }{\beta }$. Moreover, Schwarz lemma for class $\mathcal{N(\alpha )}$ is given.References
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2. Chelst, D., A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129, 3275-3278, (2001). https://doi.org/10.1090/S0002-9939-01-06144-5
3. Dubinin, V. N., The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. 122, 3623-3629, (2004). https://doi.org/10.1023/B:JOTH.0000035237.43977.39
4. Dubinin, V. N., Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207, 825-831, (2015).
https://doi.org/10.1007/s10958-015-2406-5
5. Golusin, G. M., Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Moscow 1966.
6. Jack, I. S., Functions starlike and convex of order , J. London Math. Soc. 3, 469-474, (1971). https://doi.org/10.1112/jlms/s2-3.3.469
7. Jeong, M., The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21, 275-284, (2014).
8. Jeong, M., The Schwarz lemma and boundary fixed points, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18, 219-227, (2011). https://doi.org/10.7468/jksmeb.2011.18.3.275
9. Krantz , S. G and Burns, D. M., Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Amer. Math. Soc. 7, 661-676, (1994). https://doi.org/10.2307/2152787
10. Mateljevic, M., Hyperbolic geometry and Schwarz lemma, ResearchGate 2016.
11. Mateljevic, M., Schwarz lemma, the Carath'eodory and Kobayashi Metrics and Applications in Complex Analysis, XIX GEOMETRICAL SEMINAR, At Zlatibor 1-12, (2016).
12. Mateljevic, M., Ahlfors-Schwarz lemma and curvature, Kragujevac J. Math. 25, 155-164, (2003).
13. Mateljevic, M., Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation), ResearchGate, 2015.
14. Osserman, R., A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128, 3513-3517, (2000).
https://doi.org/10.1090/S0002-9939-00-05463-0
15. Ornek, B. N., Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50, 2053-2059, (2013). https://doi.org/10.4134/BKMS.2013.50.6.2053
16. Ornek, B. N. and Akyel, T., Some results a certain class of holomorphic functions at the boundary of the unit disc, 2nd International Conference of Mathematical Sciences (ICMS 2018) 31 July-06 August 2018, Istanbul, Turkey. https://doi.org/10.1063/1.5095115
17. Pommerenke, Ch., Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
https://doi.org/10.1007/978-3-662-02770-7
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2019-10-14
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