Classes of Harmonic Functions Defined by the Carlson–Shaffer Operator

The Carlson–Shaffer operator plays an important role in the geometric theory of analytic functions. It is associated with the hypergeometric function and the incomplete beta function. The Carlson–Shaffer operator generalizes various other linear operators, such as the Ruscheweyh derivative operator, the Bernardi–Libera–Livingston operator, and the Srivastava–Owa operator. Ideas in the theory of analytic functions are often symmetrically transferred to the theory of harmonic functions. By using the Carlson–Shaffer operator, we introduce a class of harmonic functions defined by weak subordination. Next, we give some necessary and sufficient coefficient conditions for the class of functions. Furthermore, we determine coefficient estimates, distortion bounds, extreme points, and radii of starlikeness and convexity for the defined class.