Some lower bound for holomorphic functions at the boundary

Bülent Nafi Örnek

doi:10.26637/MJM0601/0019

Abstract

In this paper, a boundary version of the Schwarz lemma for classes $\mathscr{H}(\alpha)$ is investigated. For the function $f(z)=1+c_1 z+c_2 z^2+\ldots$ defined in the unit disc such that $f(z) \in \mathscr{H}(\alpha)(0<\alpha \leq 1)$, we estimate a modulus of the angular derivative of $f(z)$ function at the boundary point $b$ with $f(b)=e^{\frac{\pi \alpha}{2}}$. The sharpness of these inequalities is also proved.

Keywords:

Holomorphic function, Jack’s lemma, Angular derivative

Mathematics Subject Classification:

Mathematics

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Bülent Nafi Örnek Department of Computer Engineering, Amasya University, Merkez-Amasya 05100, Turkey.

Pages: 145-150
Date Published: 01-01-2018
Vol. 6 No. 01 (2018): Malaya Journal of Matematik (MJM)

H. P. Boas, Julius and Julia: Mastering the Art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), 770785.

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

V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disc, J. Math. Sci. 122 (2004), 3623-3629.

V. N. Dubinin, Bounded holomorphic functions covering no concentric circles, J. Math. Sci. 207 (2015), 825-831.

G. M. Golusin, Geometric Theory of Functions of Complex Variable [in Russian], 2nd edn., Moscow 1966.

I. S. Jack, Functions starlike and convex of order $alpha$, J. London Math. Soc. 3 (1971), 469-474.

M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 21 (2014), 275-284.

M. Jeong, The Schwarz, lemma and boundary fixed points, J. Korean Soc. Math. Educ. Ser. B: Pure Appl. Math. 18 (2011), 219-227.

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.

X. Tang and T. Liu, The Schwarz Lemma at the Boundary of the Egg Domain $B_{p_1, p_2}$ in $mathbb{C}^n$, Canad. Math. Bull. 58 (2015), 381-392.

X. Tang, T. Liu and J. Lu, Schwarz lemma at the boundary of the unit polydisk in $mathbb{C}^n$, Sci. China Math. 58 (2015), $1-14$.

M. Mateljević, The Lower Bound for the Modulus of the Derivatives and Jacobian of Harmonic Injective Mappings, Filomat 29:2 (2015), 221-244.

M. Mateljević, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Revue Roum. Math. Pures Appl. Vol. 51 (2006) 56, 711-722.

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

M. Mateljević, Note on Rigidity of Holomorphic Mappings & Schwarz and Jack Lemma (in preparation), ResearchGate.

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

T. Aliyev Azeroğlu and B. N. Örnek, A refined Schwarz inequality on the boundary, Complex Variables and Elliptic Equations $mathbf{5 8}$ (2013), 571-577.

B. N. Örnek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), 2053-2059.

Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.

M. Elin, F. Jacobzon, M. Levenshtein, D. Shoikhet, The Schwarz lemma: Rigidity and Dynamics, Harmonic and Complex Analysis and its Applications. Springer International Publishing, (2014), 135-230.

Taishun Liu, Jianfei Wang, Xiaomin Tang, Schwarz Lemma at the Boundary of the Unit Ball in $mathbb{C}^n$ and Its Applications, J. Geom. Anal 25 (2015), 1890-1914.

$mathrm{H}$. Unkelbach, Uber die Randverzerrung bei konformer Abbildung, Math. Z. 43 (1938), 739-742.