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.