In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.
In the present paper, we have proved the sharp inequality $|H_{3,1}(f)|$ $\le 4$ and $|H_{3,1}(f)|\le 1$ for analytic functions $f$ with $a_n:=f^{(n)}(0)/n!,\ n\in\mathbb{N},$ such that $$\mathrm{Re}\, \frac{f(z)}{z}> \alpha,\quad z\in\mathbb{D}:=\{z \in\mathbb{C} : |z|<1\}$$ for $\alpha=0$ and $\alpha=1/2,$ respectively, where \begin{equation*} H_{3,1}(f):= \begin{vmatrix} a_1 & a_2 & a_3 \\ a_2 & a_3 & a_4 \\ a_3 & a_4 & a_5 \end{vmatrix} \end{equation*} is the third Hankel determinant.
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