Estimates for λ-spirallike function of complex order on the boundary
DOI:
https://doi.org/10.37863/umzh.v74i1.2375Keywords:
Schwarz lemma on the boundary, Julia-Wolff lemma, lambda-spirallikefunctionAbstract
UDC 517.5
We give some results for λ -spirallike function of complex order at the boundary of the unit disc U. The sharpness of these results is also proved. Furthermore, three examples for our results are considered.
References
F. M. Al-Oboudi, M. M. Haidan, Spirallike functions of complex order, J. Nat. Geom., 19, no. 1-2, 53 – 72 (2000).
T. A. Azeroglu, B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Var. and Elliptic Equat., 58, no. 4, 571 – 577 (2013), https://doi.org/10.1080/17476933.2012.718338 DOI: https://doi.org/10.1080/17476933.2012.718338
H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly, 117, no. 9, 770 – 785 (2010), https://doi.org/10.4169/000298910X521643 DOI: https://doi.org/10.4169/000298910x521643
D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc., 129, no. 11, 3275 – 3278 (2001), https://doi.org/10.1090/S0002-9939-01-06144-5 DOI: https://doi.org/10.1090/S0002-9939-01-06144-5
V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci., 122, no. 6, 3623 – 3629 (2004), https://doi.org/10.1023/B:JOTH.0000035237.43977.39 DOI: https://doi.org/10.1023/B:JOTH.0000035237.43977.39
V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci., 207, no. 6, 825 – 831 (2015), https://doi.org/10.1007/s10958-015-2406-5 DOI: https://doi.org/10.1007/s10958-015-2406-5
G. M. Golusin, Geometric theory of functions of complex variable (in Russian), 2nd ed., Moscow (1966).
M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B. Pure and Appl. Math., 21 , 275 – 284 (2014), https://doi.org/10.7468/jksmeb.2014.21.3.219 DOI: https://doi.org/10.7468/jksmeb.2014.21.3.219
M. Jeong, The Schwarz lemma and boundary fixed points, J. Korean Soc. Math. Educ. Ser. B. Pure and Appl. Math., 18, 219 – 227 (2011), https://doi.org/10.7468/jksmeb.2011.18.3.275 DOI: https://doi.org/10.7468/jksmeb.2011.18.3.275
S. G. Krantz, D. M. Burns, Rigidity of holomorphic mappings and a new Schwarz Lemma at the boundary, J. Amer. Math. Soc., 7, no. 3, 661 – 676 (1994), https://doi.org/10.2307/2152787 DOI: https://doi.org/10.1090/S0894-0347-1994-1242454-2
P. R. Mercer, Sharpened versions of the Schwarz lemma, J. Math. Anal. and Appl., 205, no. 2, 508 – 511 (1997), https://doi.org/10.1006/jmaa.1997.5217 DOI: https://doi.org/10.1006/jmaa.1997.5217
P. R. Mercer, Boundary Schwarz inequalities arising from Rogosinski’s lemma, J. Class. Anal., 12, no. 2, 93 – 97 (2018), https://doi.org/10.7153/jca-2018-12-08 DOI: https://doi.org/10.7153/jca-2018-12-08
M. Mateljevi´c, The Lower Bound for the Modulus of the derivatives and Jacobian of harmonic Injective mappings, Filomat, 29, no. 2, 221 – 244 (2015), https://doi.org/10.2298/FIL1502221M DOI: https://doi.org/10.2298/FIL1502221M
M. Mateljevi´c, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roum. Math. Pures Vet Appl., 51, no. 5-6, 711 – 722 (2006).
M. Mateljevi´c, 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, no. 12, 3513 – 3517 (2000), https://doi.org/10.1090/S0002-9939-00-05463-0 DOI: https://doi.org/10.1090/S0002-9939-00-05463-0
B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc., 50, no. 6, 2053 – 2059 (2013), https://doi.org/10.4134/BKMS.2013.50.6.2053 DOI: https://doi.org/10.4134/BKMS.2013.50.6.2053
B. N. Örnek, T. Akyel, Sharpened forms of the generalized Schwarz inequality on the boundary, Proc. Indian Acad. Sci. (Math. Sci.), 126, no. 1, 69 – 78 (2016), https://doi.org/10.1007/s12044-015-0255-2 DOI: https://doi.org/10.1007/s12044-015-0255-2
Ch. Pommerenke, Boundary behaviour of conformal maps, Springer-Verlag, Berlin (1992), https://doi.org/10.1007/978-3-662-02770-7 DOI: https://doi.org/10.1007/978-3-662-02770-7
D. Shoikhet, M. Elin, F. Jacobzon, M. Levenshtein, The Schwarz lemma: rigidity and dynamics, harmonic and complex analysis and its applications, Springer Int. Publ., 135 – 230 (2014), https://doi.org/10.1007/978-3-319-01806-5_3 DOI: https://doi.org/10.1007/978-3-319-01806-5_3
