Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2016; 53(2): 325-334

Printed March 31, 2016

https://doi.org/10.4134/BKMS.2016.53.2.325

Copyright © The Korean Mathematical Society.

Inequalities for the angular derivatives of certain classes of holomorphic functions in the unit disc

B\"{u}lent Nafi \"{O}rnek

Amasya University

Abstract

In this paper, a boundary version of the Schwarz lemma is investigated. We take into consideration a function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$ holomorphic in the unit disc and $\left\vert \frac{f(z)}{\lambda f(z)+(1-\lambda )z}-\alpha \right\vert <\alpha $ for $\left\vert z\right\vert <1$, where $\frac{1}{2}<\alpha \leq \frac{1}{1+\lambda }$, $ 0\leq $ $\lambda <1$. If we know the second and the third coefficient in the expansion of the function $f(z)=z+c_{p+1}z^{p+1}+c_{p+2}z^{p+2}+\cdots$, then we can obtain more general results on the angular derivatives of certain holomorphic function on the unit disc at boundary by taking into account $ c_{p+1}$, $c_{p+2}$ and zeros of $f(z)-z$. We obtain a sharp lower bound of $ \left\vert f^{\prime }(b)\right\vert $ at the point $b$, where $\left\vert b\right\vert =1$.

Keywords: Schwarz lemma on the boundary, holomorphic function, angular derivative, Julia-Wolff-Lemma

MSC numbers: 30C80