Radius Problems for Starlike Functions Associated with the Sine Function

Let S s ∗ be the class of normalized analytic functions f defined on the unit disk such that the quantity z f ′ ( z ) / f ( z ) lies in an eight-shaped region in the right-half plane, which is the image of the unit disk under an entire function defined by φ ( z ) = 1 + sin z . For this class, we determine the S s ∗ -radii for the class of Janowski starlike functions and some other geometrically defined classes. A relation between this class and the class of Janowski starlike functions is also discussed.