The Upper Bounds of the Third Hankel Determinant for Two Subclasses of Analytic Functions

Let H denote the family of all analytic functions with the form $f(z)=z+a_2 z^2+a_3 z^3+\cdots$ in the unit disk $U=\{z:|z|<1\}$. Two subclasses of analytic functions ST_s and R(1/2) which are difined in the unit disk U are introduced, respectively, i.e., $\mathrm{ST}_{\mathrm{s}}=\left\{f ? H: \operatorname{Re} \frac{2 z f^{\prime}(z)}{f(z)-f(-z)}>0, z ? U\right\}, R\left(\frac{1}{2}\right)=\left\{f ? H: \operatorname{Re} \frac{f(z)}{z}>\frac{1}{2}, z ? U\right\}.$ And the bounds of $\left|H_{3, 1}(f)\right|$ for subfamilies of ST_s and R(1/2) are obtained.