Korean J. Math. Vol. 29 No. 4 (2021) pp.785-800
DOI: https://doi.org/10.11568/kjm.2021.29.4.785

On bounds for the derivative of analytic functions at the boundary

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Published: Dec 30, 2021
Keywords:
Analytic function, Schwarz lemma. Angular derivative, Julia-Wolff lemma.
Mathematics Subject Classification:
30C80

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Bülent Nafi Örnek
Department of Computer Engineering, Amasya University Merkez-Amasya, 05100, Turkey
Tuğba Akyel
The Faculty of Engineering and Natural Sciences, Maltepe University Maltepe, Istanbul, Turkey

Abstract

In this paper, we obtain a new boundary version of the Schwarz lemma for analytic function. We give sharp upper bounds for |f′(0)| and sharp lower bounds for |f′(c)| with c∈∂D={z:|z|=1}. Thus we present some new inequalities for analytic functions. Also, we estimate the modulus of the angular derivative of the function f(z) from below according to the second Taylor coefficients of f about z=0 and z=z0≠0. Thanks to these inequalities, we see the relation between |f′(0)| and ℜf(0). Similarly, we see the relation between ℜf(0) and |f′(c)| for some c∈∂D. The sharpness of these inequalities is also proved.



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