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

Commun. Korean Math. Soc. 2021 Vol. 36, No. 4, 743-757 https://doi.org/10.4134/CKMS.c200336 Published online May 13, 2021 Printed October 31, 2021

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.