Radius problems for functions associated with a nephroid domain

Let S Ne ∗ be the collection of all analytic functions f ( z ) defined on the open unit disk D and satisfying the normalizations f ( 0 ) = f ′ ( 0 ) - 1 = 0 such that the quantity z f ′ ( z ) / f ( z ) assumes values from the range of the function φ Ne ( z ) : = 1 + z - z 3 / 3 , z ∈ D , which is the interior of the nephroid given by ( u - 1 ) 2 + v 2 - 4 9 3 - 4 v 2 3 = 0 . In this work, we find sharp S Ne ∗ -radii for several geometrically defined function classes introduced in the recent past. In particular, S Ne ∗ -radius for the starlike class S ∗ is found to be 1/4. Moreover, radii problems related to the families defined in terms of ratio of functions are also discussed. Sharpness of certain radii estimates are illustrated graphically.