Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

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Commun. Korean Math. Soc. 2021; 36(4): 743-757

Online first article May 13, 2021      Printed October 31, 2021

https://doi.org/10.4134/CKMS.c200336

Copyright © The Korean Mathematical Society.

Some remarks for $\lambda$-spirallike function of complex order at the boundary of the unit disc

Tu\u{g}ba Akyel

Maltepe University

Abstract

We consider a different version of Schwarz Lemma for $\lambda$-spirallike function of complex order at the boundary of the unit disc $D.$ We estimate the modulus of the angular derivative of the function $\frac{zf'(z)}{f(z)}$ from below for $\lambda$-spirallike function $f(z)$ of complex order at the boundary of the unit disc $D$ by taking into account the zeros of the function $f(z)-z$ which are different from zero. We also estimate the same function with the second derivatives of the function $f$ at the points $z=0$ and $z=z_0\neq 0.$ We show the sharpness of these estimates and present examples.

Keywords: Angular derivative, holomorphic function, $\lambda $-spirallike function, Schwarz lemma

MSC numbers: Primary 30C80, 32A10