Estimates concerned with Hankel determinant for M(α) class

Abstract

In this paper, we give an upper bounds of Hankel determinant of ($H_{2}(1))$for the classes of $\mathcal{M}\left( \mathcal{\alpha }\right) $, $\alpha\in %TCIMACRO{\U{2102} }%%BeginExpansion\mathbb{C}%EndExpansion$. Also, for $\mathcal{M}\left( \mathcal{\alpha }\right) $, we obtain sharpestimate for the classical Fekete-Szeg\"{o} inequality. That is, we will geta sharp upper bound for the Hankel determinant $H_{2}(1)=c_{3}-c_{2}^{2}$.Moreover, in a class of analytic functions on the unit disc, assuming theexistence of angular limit on the boundary point, the estimations below ofthe modulus of angular derivative have been obtained.