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

Lemma 2.1. If $p \in \mathcal{P}$ is of the form (1.6) with $c_1 \geq 0$ , then $$2c_2 = c_1^2 + (4 - c_1^2)\zeta, (2.1)$$ $$4c_3 = c_1^3 + (4 - c_1^2)c_1\zeta(2 - \zeta) + 2(4 - c_1^2)(1 - |\zeta|^2)\eta$$ (2.2) and $$8c_4 = c_1^4 + (4 - c_1^2)\zeta \left[ c_1^2(\zeta^2 - 3\zeta + 3) + 4\zeta \right]$$ $$-4(4 - c_1^2)(1 - |\zeta|^2) \left[ c_1(\zeta - 1)\eta + \overline{\zeta}\eta^2 - \left(1 - |\eta|^2\right)\xi \right]$$ (2.3) for some $\zeta, \eta, \xi \in \overline{\mathbb{D}} := \{z \in \mathbb{C} : |z| \le 1\}.$ We will now estimate the third order Hankel determinant $H_{3,1}(f)$ for $f \in \mathcal{S}^*(1/2)$ .