We give some results obtained for λ-spirallike functions of complex order on the boundary of the unit disc U. The sharpness of these results is also proved. Furthermore, three examples of our results are considered.
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F. M. Al-Oboudi and M. M. Haidan, “Spirallike functions of complex order,” J. Nat. Geom., 19, 53–72 (2000).
T. A. Azeroğlu and B. N. Örnek, “A refined Schwarz inequality on the boundary,” Complex Var. Elliptic Equat., 58, 571–577 (2013).
H. P. Boas, “Julius and Julia: mastering the art of the Schwarz lemma,” Amer. Math. Monthly, 117, 770–785 (2010).
D. Chelst, “A generalized Schwarz lemma at the boundary,” Proc. Amer. Math. Soc., 129, 3275–3278 (2001).
V. N. Dubinin, “The Schwarz inequality on the boundary for functions regular in the disc,” J. Math. Sci. (N.Y.), 122, 3623–3629 (2004).
V. N. Dubinin, “Bounded holomorphic functions covering no concentric circles,” J. Math. Sci. (N.Y.), 207, 825–831 (2015).
G. M. Golusin, Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Nauka, Moscow (1966).
M. Jeong, “The Schwarz lemma and its applications at a boundary point,” J. Korean Soc. Math. Educ., Ser. B. Pure Appl. Math., 21, 275–284 (2014).
M. Jeong, “The Schwarz lemma and boundary fixed points,” J. Korean Soc. Math. Educ., Ser. B. Pure Appl. Math., 18, 219–227 (2011).
S. G. Krantz and D. M. Burns, “Rigidity of holomorphic mappings and a new Schwarz lemma at the boundary,” J. Amer. Math. Soc., 7, 661–676 (1994).
P. R. Mercer, “Sharpened versions of the Schwarz lemma,” J. Math. Anal. Appl., 205, No. 2, 508–511 (1997).
P. R. Mercer, “Boundary Schwarz inequalities arising from Rogosinski’s lemma,” J. Class. Anal., 12, 93–97 (2018).
M. Mateljević, “The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings,” Filomat, 29, 221–244 (2015).
M. Mateljević, “Distortion of harmonic functions and harmonic quasiconformal quasi-isometry,” Rev. Roumaine Math. Pures Appl., 51, No. 5-6, 711–722 (2006).
M. Mateljević, “Ahlfors–Schwarz lemma and curvature,” Kragujevac J. Math., 25, 155–164 (2003).
R. Osserman, “A sharp Schwarz inequality on the boundary,” Proc. Amer. Math. Soc., 128, 3513–3517 (2000).
B. N. Örnek, “Sharpened forms of the Schwarz lemma on the boundary,” Bull. Korean Math. Soc., 50, No. 6, 2053–2059 (2013).
B. N. Örnek and T. Akyel, “Sharpened forms of the generalized Schwarz inequality on the boundary,” Proc. Indian Acad. Sci. Math. Sci, 126, No. 1, 69–78 (2016).
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin (1992).
D. Shoikhet, M. Elin, F. Jacobzon, and M. Levenshtein, The Schwarz Lemma: Rigidity and Dynamics. Harmonic and Complex Analysis and Its Applications, in: Trends Math., Birkhäuser/Springer, Cham (2014), pp. 135–230.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 1, pp. 3–13, January, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i1.2375.
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Akyel, T. Estimates for λ-Spirallike Functions of Complex Order on the Boundary. Ukr Math J 74, 1–14 (2022). https://doi.org/10.1007/s11253-022-02043-z
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DOI: https://doi.org/10.1007/s11253-022-02043-z