# INEQUALITIES FOR THE NON-TANGENTIAL DERIVATIVE AT THE BOUNDARY FOR HOLOMORPHIC FUNCTION

• Ornek, Bulent Nafi (Department of Mathematics Gebze Institute of Technology)
• Published : 2014.07.31
• 107 25

#### Abstract

In this paper, we present some inequalities for the non-tangential derivative of f(z). For the function $f(z)=z+b_{p+1}z^{p+1}+b_{p+2}z^{p+2}+{\cdots}$ defined in the unit disc, with ${\Re}$\frac{f^{\prime}(z)}{{\lambda}f{\prime}(z)+1-{\lambda}}$$ > ${\beta}$, $0{\leq}{\beta}$ < 1, $0{\leq}{\lambda}$ < 1, we estimate a module of a second non-tangential derivative of f(z) function at the boundary point ${\xi}$, by taking into account their first nonzero two Maclaurin coefficients. The sharpness of these estimates is also proved.

#### Keywords

Schwarz lemma on the boundary;holomorphic function;second non-tangential derivative;critical points

#### References

1. T. Aliyev Azeroglu and B. N. Ornek, A refined Schwarz inequality on the boundary, Complex Var. Elliptic Equ. 58 (2013), no. 4, 571-577. https://doi.org/10.1080/17476933.2012.718338
2. H. P. Boas, Julius and Julia: Mastering the art of the Schwarz lemma, Amer. Math. Monthly 117 (2010), no. 9, 770-785. https://doi.org/10.4169/000298910X521643
3. V. N. Dubinin, The Schwarz inequality on the boundary for functions regular in the disk, J. Math. Sci. 122 (2004), no. 6, 3623-3629. https://doi.org/10.1023/B:JOTH.0000035237.43977.39
4. G. M. Golusin, Geometric Theory of Functions of Complex Variable, [in Russian], 2nd edn., Moscow, 1966.
5. H. T. Kaptanoglu, Some refined Schwarz-Pick lemmas, Michigan Math. J. 50 (2002), no. 3, 649-664. https://doi.org/10.1307/mmj/1039029986
6. B. N. Ornek, Sharpened forms of the Schwarz lemma on the boundary, Bull. Korean Math. Soc. 50 (2013), no. 6, 2053-2059. https://doi.org/10.4134/BKMS.2013.50.6.2053
7. R. Osserman, A sharp Schwarz inequality on the boundary, Proc. Amer. Math. Soc. 128 (2000), no. 12, 3513-3517. https://doi.org/10.1090/S0002-9939-00-05463-0
8. Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Springer-Verlag, Berlin, 1992.
9. H. Unkelbach, Uber die Randverzerrung bei konformer Abbildung, Math. Z. 43 (1938), no. 1, 739-742. https://doi.org/10.1007/BF01181115