Sharpened forms of analytic functions concerned with Hankel determinant

Abstract

In this paper, we present a Schwarz lemma at the boundary for analytic functions at the unit disc, which generalizes classical Schwarz lemma for bounded analytic functions. For new inequalities, the results of Jack's lemma and Hankel determinant were used. We will get a sharp upper bound for Hankel determinant $H_{2}(1)$. Also, 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.