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., \mathrmST_\mathrms=\left\f ? H: \operatornameRe \frac2 z f^\prime(z)f(z)-f(-z)>0, z ? U\right\, R\left(\frac12\right)=\left\f ? H: \operatornameRe \fracf(z)z>\frac12, z ? U\right\. And the bounds of \left|H_3, 1(f)\right| for subfamilies of ST_s and R(1/2) are obtained.