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http://dx.doi.org/10.4134/CKMS.c180469

SOME REMARKS OF THE CARATHÉODORY'S INEQUALITY ON THE RIGHT HALF PLANE  

Ornek, Bulent Nafi (Department of Computer Engineering Amasya University)
Publication Information
Communications of the Korean Mathematical Society / v.35, no.1, 2020 , pp. 201-215 More about this Journal
Abstract
In this paper, a boundary version of Carathéodory's inequality on the right half plane for p-valent is investigated. Let Z(s) = 1+cp (s - 1)p +cp+1 (s - 1)p+1 +⋯ be an analytic function in the right half plane with ℜZ(s) ≤ A (A > 1) for ℜs ≥ 0. We derive inequalities for the modulus of Z(s) function, |Z'(0)|, by assuming the Z(s) function is also analytic at the boundary point s = 0 on the imaginary axis and finally, the sharpness of these inequalities is proved.
Keywords
Schwarz lemma on the boundary; $Carath{\acute{e}}odory^{\prime}s$ inequality; analytic function;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 T. Aliyev Azeroglu and B. N. Ornek, A refined Schwarz inequality on the boundary, Complex Var. Elliptic Equ. 58 (2013), no. 4, 571-577. https://doi.org/10.1080/17476933.2012.718338   DOI
2 H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), no. 9, 770-785. https://doi.org/10.4169/000298910X521643   DOI
3 D. M. Burns and S. G. Krantz, Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary, J. Amer. Math. Soc. 7 (1994), no. 3, 661-676. https://doi.org/10.2307/2152787   DOI
4 D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3275-3278. https://doi.org/10.1090/S0002-9939-01-06144-5   DOI
5 V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. (N.Y.) 122 (2004), no. 6, 3623-3629. https://doi.org/10.1023/B:JOTH.0000035237.43977.39   DOI
6 G. M. Goluzin, Geometric Theory of Functions of Complex Variable, (Russian), Second edition. Edited by V. I. Smirnov. With a supplement by N. A. Lebedev, G. V. Kuzmina and Ju. E. Alenicyn, Izdat. "Nauka", Moscow, 1966.
7 G. Kresin and V. Maz'ya, Sharp real-part theorems, translated from the Russian and edited by T. Shaposhnikova, Lecture Notes in Mathematics, 1903, Springer, Berlin, 2007.
8 M. Mateljevi'c, Rigidity of holomorphic mappings & Schwarz and Jack lemma, DOI:10.13140/RG.2.2.34140.90249, In press.
9 P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. Appl. 205 (1997), no. 2, 508-511. https://doi.org/10.1006/jmaa.1997.5217   DOI
10 P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski's lemma, J. Class. Anal. 12 (2018), no. 2, 93-97. https://doi.org/10.7153/jca-2018-12-08   DOI
11 P. R. Mercer, An improved Schwarz lemma at the boundary, Open Math. 16 (2018), no. 1, 1140-1144. https://doi.org/10.1515/math-2018-0096   DOI
12 B. N. Ornek, Some results of the Caratheodory's inequality at the boundary, Commun. Korean Math. Soc. 33 (2018), no. 4, 1205-1215. https://doi.org/10.4134/CKMS.c170372   DOI
13 R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0   DOI
14 B. N. Ornek, Caratheodory's inequality on the boundary, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 22 (2015), no. 2, 169-178.
15 B. N. Ornek, A sharp Caratheodory's inequality on the boundary, Commun. Korean Math. Soc. 31 (2016), no. 3, 533-547. https://doi.org/10.4134/CKMS.c150194   DOI
16 B. N. Ornek and T. Duzenli, Schwarz lemma for driving point impedance functions and its circuit applications, Int J. Circ. Theor Appl. 47 (2019), 813-824. https://doi.org/10.1002/cta.2616   DOI