The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2
The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2
In the present paper, we proved the sharp inequality $$|H_{3,1}(f)|\le 1/9$$ for analytic functions f with $$a_n:=f^{(n)}(0)/n!,\ n\in {\mathbb {N}},\ a_1:=1,$$ such that $$\begin{aligned} {{\mathrm{Re}}}\frac{zf'(z)}{f(z)}> \frac{1}{2},\quad z\in {\mathbb {D}}:=\{z \in {\mathbb {C}} : |z|<1\}, \end{aligned}$$ where $$\begin{aligned} H_{3,1}(f):= \begin{vmatrix} a_1&\quad a_2&\quad a_3 \\ a_2&\quad a_3&\quad a_4 \\ a_3&\quad a_4&\quad a_5 …