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 |H3,1(f)|≤ 1/9 for analytic functions f with aₙ:=f⁽ⁿ⁾(0)/n!,\ n∈ N,\ a₁:=1, such that aligned Rezf'(z)/f(z)> 1/2, z∈ D:= ∈ C : |z|<1\, aligned where aligned H3,1(f):= vmatrix a₁& a₂& a₃ \\ a₂& a₃& a₄ \\ a₃& a₄& a₅ vmatrix aligned is the third Hankel determinant.