On harmonic functions defined by derivative operator

Abstract

Let ℋ denote the class of functions f=h+ ḡ that are harmonic univalent and sense-preserv- ing in the unit disk U=z:|z|<1, where h(z)=z+ ∑ k=2 ∞ ak zk, g(z)= ∑ k=1 ∞ bk zk ( | b1 |<1) . In this paper, we introduce the class Mℋ ( n,λ,α) of functions f=h+ ḡ which are harmonic in U. A sufficient coefficient of this class is determined. It is shown that this coefficient bound is also necessary for the class M ℋ̄ ( n,λ,α) if fn (z)=h+ gn ̄ ∈Mℋ ( n,λ,α), where h(z)=z- ∑ k=2 ∞ | ak | zk, gn (z)=( -1)n ∑ k=1 ∞ | bk |zk and n∈0 . Coefficient conditions, such as distortion bounds, convolution conditions, convex combination, extreme points, and neighborhood for the class M ℋ̄ ( n,λ,α), are obtained.