Harmonic mappings related to the m-fold starlike functions

In the present paper we will give some properties of the subclass of harmonic mappings which is related to m-fold starlike functions in the open unit disc D={z||z|<1}. Throughout this paper we restrict ourselves to the study of sense-preserving harmonic mappings. We also note that an elegant and complete treatment theory of the harmonic mapping is given in Durens monograph (Duren, 1983). The main aim of us to investigate some properties of the new class of us which represented as in the following form,S∗H(m)=f=h(z)+g(z)‾|f∈SH(m),g′(z)h′(z)≺b1p(z),h(z)∈S∗(m),p(z)∈P(m),where h(z)=z+∑n=1∞amn+1zmn+1, g(z)=∑n=0∞bmn+1zmn+1,|b1|<1.