A SHARP CARATHÉODORY'S INEQUALITY ON THE BOUNDARY

Title & Authors
A SHARP CARATHÉODORY'S INEQUALITY ON THE BOUNDARY
Ornek, Bulent Nafi;

Abstract
In this paper, a generalized boundary version of $\small{Carath{\acute{e}}odory^{\prime}s}$ inequality for holomorphic function satisfying $\small{f(z)= f(0)+a_pz^p+{\cdots}}$, and $\small{{\Re}f(z){\leq}A}$ for $\small{{\mid}z{\mid}}$<1 is investigated. Also, we obtain sharp lower bounds on the angular derivative $\small{f^{\prime}(c)}$ at the point c with $\small{{\Re}f(c)=A}$. The sharpness of these estimates is also proved.
Keywords
holomorphic function;Schwarz lemma on the boundary;$\small{Carath{\acute{e}}odory^{\prime}s}$ inequality;
Language
English
Cited by
References
1.
T. Akyel and B. N. Ornek, A sharp Schwarz lemma at the boundary, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 22 (2015), no. 3, 263-273.

2.
T. Aliyev Azeroglu and B. N. Ornek, A refined Schwarz inequality on the boundary, Complex Var. Elliptic Equ. 58 (2013), no. 4, 571-577.

3.
H. P. Boas, Julius and Julia: Mastering the art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), no. 9, 770-785.

4.
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), no. 3, 661-676.

5.
D. Chelst, A generalized Schwarz lemma at the boundary, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3275-3278.

6.
V. N. Dubinin, On the Schwarz inequality on the boundary for functions regular in the disk, J. Math. Sci. 122 (2004), no. 6, 3623-3629.

7.
G. M. Golusin, Geometric Theory of Functions of Complex Variable, 2nd edn., Moscow 1966.

8.
M. Jeong, The Schwarz lemma and boundary fixed points, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 18 (2011), no. 3, 275-284.

9.
M. Jeong, The Schwarz lemma and its applications at a boundary point, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 21 (2014), no. 3, 219-227.

10.
G. Kresin and V. Maz'ya, Sharp real-part theorems. A unified approach, Translated from the Russian and edited by T. Shaposhnikova. Lecture Notes in Mathematics, 1903. Springer, Berlin, 2007.

11.
M. Mateljevic, Ahlfors-Schwarz lemma and curvature, Kragujevac J. Math. 25 (2003), 155-164.

12.
M. Mateljevic, Distortion of harmonic functions and harmonic quasiconformal quasi-isometry, Rev. Roumaine Math. Pures Appl. 51 (2006), no. 5-6, 711-722.

13.
M. Mateljevic, The lower bound for the modulus of the derivatives and Jacobian of harmonic injective mappings, Filomat 29 (2015), no. 2, 221-244.

14.
M. Mateljevic, Note on rigidity of holomorphic mappings & Schwarz and Jack lemma, (in preparation) ResearchGate.

15.
B. N. Ornek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), no. 6, 2053-2059.

16.
B. N. Ornek, Caratheodory's inequality on the boundary, J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math. 22 (2015), no. 2, 169-178.

17.
R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3513-3517.

18.
Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.

19.
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), no. 8, 1639-1652.