On a subclass of harmonic close-to-convex mappings

Let H denote the class of harmonic functions f defined in D : = { z ∈ C : | z | < 1 } , and normalized by f ( 0 ) = 0 = f z ( 0 ) - 1 . In this paper, for α ≥ 0 , we consider the subclass W H 0 ( α ) of H , defined by W H 0 ( α ) : = f = h + g ¯ ∈ H : Re ( h ′ ( z ) + α z h ′ ′ ( z ) ) > | g ′ ( z ) + α z g ′ ′ ( z ) | , z ∈ D . For f ∈ W H 0 ( α ) , we prove the Clunie–Sheil-Small coefficient conjecture, and give some growth, convolution, and convex combination theorems. We also determine the value of r so that the partial sums of functions in W H 0 ( α ) are close-to-convex in | z | < r .