SOME RESULTS CONCERNED WITH HANKEL DETERMINANT FOR N(α) CLASS
Commun. Korean Math. Soc.
Published online May 13, 2021
Bülent Nafi ÖRNEK and Gizem ATLI
Amasya University; Amasya University
Abstract : In this paper, we give some results an upper bounds of Hankel determinant of
$H_{2}(1)$ for the classes of $N\left( \alpha \right) =\left( \frac{z}{f(z)}%
\right) ^{2}f^{\prime }(z)-\alpha $, $\alpha \in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
$. We get a sharp upper bound for $H_{2}(1)=c_{3}-c_{2}^{2}$ for $\mathcal{N}%
\left( \mathcal{\alpha }\right) $ by adding $z_{1},z_{2},...,z_{n}$ 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 : 30C80, 32A10
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