Estimates for $\lambda$-spirallike function of complex order on the boundary

T.  Akyel

doi:10.37863/umzh.v74i1.2375

T. Akyel Maltepe Univ., Istanbul, Turkey

DOI: https://doi.org/10.37863/umzh.v74i1.2375

Keywords: Schwarz lemma on the boundary, Julia-Wolff lemma, lambda-spirallikefunction

Abstract

UDC 517.5

We give some results for $\lambda$ -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