Some results concerned with Hankel determinant for $\mathcal{N}\left( \mathcal{\alpha }\right)$ class
Commun. Korean Math. Soc. 2021 Vol. 36, No. 4, 715-727
https://doi.org/10.4134/CKMS.c200313
Published online May 13, 2021
Printed October 31, 2021
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
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