Some Estimates for Holomorphic Functions at the Boundary of the Unit Disc


Download PDF

Authors: B. N. ORNEK

DOI: 10.46793/KgJMat2003.475O

Abstract:

In this paper, for holomorphic function f(z) = z + c2z2 + c3z3 + ⋅⋅⋅ belong to the class of N(λ), it has been estimated from below the modulus of the angular derivative of the function zf′(z)-
f(z) on the boundary point of the unit disc.

Keywords:

Schwarz lemma, holomorphic function, angular limit.

References:

[1]   T. A. Azeroğlu and B. Örnek, A refined schwarz inequality on the boundary, Complex Var. Elliptic Equ. 58 (2013), 571–577.

[2]   H. P. Boas, Julius and Julia: mastering the art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), 770–785.

[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), 661–676.

[4]   D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), 3275–3278.

[5]   V. Dubinin, The Sschwarz inequality on the boundary for functions regular in the disk, J. Math. Sci. 122 (2004), 3623–3629.

[6]   V. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207(6) (2015), 825–831.

[7]   M. Elin, F. Jacobzon, M. Levenshtein and D. Shoikhet, The Schwarz lemma: rigidity and dynamics, in: Harmonic and Complex Analysis and its Applications, Springer, Switzerland, Basel, 2014, 135–230.

[8]   G. M. Goluzin, Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, Rhode Island, 1969.

[9]   M. Jeong, The Schwarz lemma and its application at a boundary point, Pure Appl. Math. 21 (2014), 219–227.

[10]   M.-J. Jeong, The Schwarz lemma and boundary fixed points, Pure Appl. Math. 18 (2011), 275–284.

[11]   M. Mateljević, Note on rigidity of holomorphic mappings & Schwarz and Jack lemma, Filomat, (to appear).

[12]   M. Mateljević, Ahlfors-Schwarz lemma and curvature, Kragujevac J. Math. 25 (2003), 155–164.

[13]   M. Mateljevic, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roumaine Math. Pures Appl. 51 (2006), 711–722.

[14]   M. Mateljević, The lower bound for the modulus of the derivatives and jacobian of harmonic injective mappings, Filomat 29 (2015), 221–244.

[15]   B. Ornek, Estimates for holomorphic functions concerned with Jack’s lemma, Publ. Inst. Math. (Beograd) (N.S.) 104(118) (2018), 231–240.

[16]   B. N. Ornek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), 2053–2059.

[17]   R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), 3513–3517.

[18]   C. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299, Springer-Verlag, Berlin, Heidelberg, 1992.

[19]   X. Tang and T. Liu, The Schwarz lemma at the boundary of the egg domain Bp1,p2 in n, Canad. Math. Bull. 58 (2015), 381–392.

[20]   X. Tang, T. Liu and J. Lu, Schwarz lemma at the boundary of the unit polydisk in n, Sci. China Math. 58 (2015), 1639–1652.

[21]   H. Unkelbach, Über die randverzerrung bei konformer abbildung, Math. Z. 43 (1938), 739–742.