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http://dx.doi.org/10.11568/kjm.2021.29.4.785

ON BOUNDS FOR THE DERIVATIVE OF ANALYTIC FUNCTIONS AT THE BOUNDARY  

Ornek, Bulent Nafi (Department of Computer Engineering, Amasya University)
Akyel, Tugba (The Faculty of Engineering and Natural Sciences, Maltepe University)
Publication Information
Korean Journal of Mathematics / v.29, no.4, 2021 , pp. 785-800 More about this Journal
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.
Keywords
Analytic function; Schwarz lemma; Angular derivative; Julia-Wolff lemma;
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