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): 715-727

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

https://doi.org/10.4134/CKMS.c200313

Copyright © The Korean Mathematical Society.

Some results concerned with Hankel determinant for $\mathcal{N}\left( \mathcal{\alpha }\right)$ class

Gizem Atli, B\"{u}lent Nafi \"{O}rnek

Amasya University; Amasya University

Abstract

In this paper, we give some results an upper bound of Hankel determinant of $H_{2}(1)$ for the classes of $\mathcal{N}\left( \mathcal{\alpha }\right) $. We get a sharp upper bound for $H_{2}(1)=c_{3}-c_{2}^{2}$ for $\mathcal{N}\left( \mathcal{\alpha }\right) $ by adding $z_{1},z_{2},\ldots,z_{n}$ zeros of $f(z)$ which are different than zero. Moreover, in a class of analytic functions on the unit disc, assuming the existence of angular limit on the boundary point, the estimations below of the modulus of angular derivative have been obtained. Finally, the sharpness of the inequalities obtained in the presented theorems are proved.

Keywords: Fekete-Szeg\"{o} functional, Julia-Wolff lemma, Hankel determinant, analytic function, Schwarz lemma, angular derivative

MSC numbers: Primary 30C80, 32A10