The Sharp Bound of the Hankel Determinant of the Third Kind for Starlike Functions of Order 1 / 2

By Adam Lecko, Young Jae Sim, B. Śmiarowska. Complex Analysis and Operator Theory, 2018

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In the present paper, we proved the sharp inequality $$|H_{3,1}(f)|\le 1/9$$|H3,1(f)|≤1/9 for analytic functions f with $$a_n:=f^{(n)}(0)/n!,\ n\in {\mathbb {N}},\ a_1:=1,$$an:=f(n)(0)/n!,n∈N,a1:=1, such that $$\begin{aligned} {{\mathrm{Re}}}\frac{zf'(z)}{f(z)}> \frac{1}{2},\quad z\in {\mathbb {D}}:=\{z \in {\mathbb {C}} : |z|<1\}, \end{aligned}$$Rezf′(z)f(z)>12,z∈D:={z∈C:|z|<1},where $$\begin{aligned} H_{3,1}(f):= \begin{vmatrix} a_1&\quad a_2&\quad a_3 \\ a_2&\quad a_3&\quad a_4 \\ a_3&\quad a_4&\quad a_5 \end{vmatrix} \end{aligned}$$H3,1(f):=a1a2a3a2a3a4a3a4a5is the third Hankel determinant.

Subfields: Geometry and Topology, Materials Chemistry, Polymers and Plastics

View Paper: https://www.semanticscholar.org/paper/f149e2e6a907b6a3c62c2d4e3ce8b80365b2c825